This friendly sideswipe, and indeed the award for Doran, Boyd's soon-to-be successor, marked an important rapprochement between the Globe and the RSC, which had initially poured scorn on the very impertinence of Sam Wanamaker's vision and determination in claiming responsibility for the Bard in London.
Now, of course, the RSC has seriously lost ground to the Globe in the capital, and it will be Doran's job to reclaim bragging rights at some time in his RSC artistic directorship.
Dromgoole was joined on the stage by his chief executive Neil Constable, who revealed that Doran had taken early inspiration from Wanamaker as an undergraduate Shakespearean, and by his education director Patrick Spottiswoode, who's been there since the start, and who recalled the imperishable cry of a Southwark councillor, irreconcilably opposed to the building of the theatre: "What's Shakespeare ever done for Southwark?"
The packed audience, many of whom it may safely be assumed had never even heard of Sam Wanamaker, erupted in a roar similar to that with which they had earlier greeted Jamie Parker's shout of "Cry God for Harry, England and St George."
For although Dromgoole's production is not a testosterone-fuelled war machine like Ed Hall's for his Propeller revival (coming to Hampstead Theatre next month) - and indeed Jamie Parker finds an unusual amount of light and shade in his portrait of the king - the show certainly catches the current Diamond Jubileee mood, the sudden surge of optimism over the national football team, the cautious, but growing, sense of pride over what London might show the world in the Olympic Games.
What I love about Globe matinees is the low level drone of the river hubbub outside as the play continues, the sense that the theatre is linked in a direct physical way to the progress of the people's day on the street, as it must have been in Shakespeare's own time.
At the end of this month, the BBC will screen Sam Mendes' The Hollow Crown, the Prince Hal tetralogy of Shakespeare histories with Ben Whishaw as Richard II, Jeremy Irons as Henry IV and Tom Hiddleston as Prince Hal/Henry V (not forgetting Simon Russell Beale as Falstaff) and I can hardly wait to see these great plays at a series of Press previews this week; they were screened years ago by the BBC in their Age of Kings series, and Orson Welles' Chimes at Midnight is a marvellous conflation of the middle two.
And then, of course, in late 2013, we shall have Jude Law as Henry V in Michael Grandage's West End occupation of the Noel Coward Theatre. The best RSC Henrys in recent memory were Kenneth Branagh (newly knighted, too, in the Queen's Birthday honours) and Michael Sheen.
But the greatest was undoubtedly Alan Howard, who was the first of the modern tortured war-mongers, with a voice that could drown a peal of bells and make the cannon boom in an astonishing and even subtly reflective manner. He was also granted, by his regular RSC director, Terry Hands, the special privilege of head-to-toe black leather and a constant follow spot.
Howard played all the kings for the RSC over two decades, but his Henry V was always my favourite. He was an actor we tended to take, and still take, for granted, and his stage appearances these days are so rare - Tiresias in Oedipus at the National, Sir Peter Teazle in The School for Scandal at the Barbican - that they have acquired the value of gold dust in the enveloping gloom.
So there was no way I was going to miss his participation in this year's wonderful Spitalfields Summer Music Festival (running till Saturday at venues all over Shoreditch and Hoxton, too) reciting Beckett fragments to the accompaniment of The Sixteen - just eight of them, as it turned out - singing a beautiful wordless score by Alec Roth, conducted by Harry Christophers.
Old Earth by Beckett - the title is one of eight fragments, or "fizzles" (meaning silent exhalations from the fundament) that Beckett wrote over 20 years and to which the American artist Jasper Johns famously attached some art work for a special edition in 1976 - gave us a decrepit, but still unlined, baby-faced Howard, leaning on a stick, staring into the abyss of his own devastated humanity on a stage of gnarled roots and tangled branches in the atmospheric, brick mausoleum of the Village Underground in Holywell Lane.
Howard performed four of the fizzles, stitching them into Roth's soundscape of cries, "wahs", chants and baby noises, aghast and moved by his own witness of himself and another person (who might have been himself after all ), dead before birth, slung in a ditch, listening at one point to his own (brilliantly recorded) distant voice like Krapp in the countryside and finally evincing the slightest hint of a beatific smile, finding peace, head bowed, in other skies, another body.
Each phrase is held by Howard to the light like a glinting sixpence he's found in the dungheap and yet this great Shakespearean can expose them all at once in an unbroken, glittering line of perfectly-phrased currency. A shard of meaning sparkles and is gone. Existence is trapped in a moment. And Howard, like anyone speaking Beckett with a true sense of music, slides imperceptibly into an Irish intonation.
One of Howard's finest post-RSC solo performances was in Christopher Logue's War Poetry. This new performance is comparable in skill and range, but it's quieter, more resigned and more deeply moving, grabbing after life and accepting the inevitable. Sirs Ian McKellen, Derek Jacobi and now Kenneth Branagh are more celebrated, and more knighted, than Alan Howard. But none of them is a greater actor.