No reads here. Is it better to call the flop min raise, and keep the pot small?

My intention with raising the flop was get called by a pair of queens or raised by a set. On the turn, I figured a pair would call a small bet and a boat would reraise, although am not sure about this logic.

I'm thinking I should've just called the flop. I coulda played this better, right???!!!!

PokerStars No-Limit Hold'em, $0.25 BB (9 handed) Hand History Converter Tool from FlopTurnRiver.com (Format: FlopTurnRiver)

saw flop|saw showdown

BB ($74.50)
UTG ($24.65)
Hero ($28.65)
MP1 ($14.30)
MP2 ($11.45)
MP3 ($25.25)
CO ($9.75)
Button ($11.70)
SB ($9.05)

Preflop: Hero is UTG+1 with A, A.
1 fold, Hero raises to $1, 2 folds, MP3 calls $1, 4 folds.

Flop: ($2.35) 4, 7, Q (2 players)
Hero bets $1.75, MP3 raises to $3.5, Hero raises to $7.75, MP3 calls $4.25.

Turn: ($17.85) 4 (2 players)
Hero bets $5, MP3 raises to $16.5, Hero folds.

Final Pot: $39.35