Ciao UomoGrasso,
June is right around the corner!
I have the Amici degli Uffizi Card... when I went recently to the Uffizi it was around 11am, they handed me a ticket that said 9am entry... so I am thinking they print out the total number of tickets for the day and hand them out - that is their way to keep track of how many each day.
Head to the front of door 2 and show the card, you'll get immediate access to the line waiting to go through the metal detectors. You might have a very short wait to pick up your "free" tickets then, but it has always been quite relative to the amount of people going in.... took me a while on a Saturday, to get from there all the way up to the 2nd floor where they ask for tickets (maybe 30 min total). There was a line after the tickets, just from people waiting to get in at the last turnstill... I'd say if you go on any other day, wait will be shorter.