Poker Stars $4.00+$0.40 No Limit Hold'em Tournament - t125/t250 Blinds + t25 - 8 players
The Official 2+2 Hand Converter Powered By DeucesCracked.com

UTG: t31663 M = 55.07
UTG+1: t1480 M = 2.57
MP1: t8680 M = 15.10
MP2: t5194 M = 9.03
Hero (CO): t7600 M = 13.22
BTN: t1600 M = 2.78
SB: t3310 M = 5.76
BB: t2350 M = 4.09

Pre Flop: (t575) Hero is CO with Q Q
UTG calls t250, 3 folds, Hero raises to t1250, 3 folds, UTG calls t1000

Flop: (t3075) T 3 T (2 players)
UTG checks, Hero bets t3000, UTG calls t3000

Turn: (t9075) K (2 players)
UTG bets t3250, Hero calls t3250

River: (t15575) 9 (2 players)
UTG bets t250, Hero calls t75 all in

I know this is played horribly after the flop, I was super tilted and didn't feel like short-stacking. My question involves the flop.

Obviously there is a lot of action around paired flops and I was unsure as to how I could extract the most value, especially from a big-stack.

Should I shove this flop 100%?

I think your line fine, it's just a bad turn card plus that I guess it hit him. Unlucky. Doesn't mean you should overshove flop, actually I like betting a little less like 2.5k so he might assume more FE on a c/r.