4 turns:
T1: buy Doctor for 6, overpaying by 3, trashing 3 Estates.
T2: DCCCC, trash 3 Coppers, buy Rats.
T3: RCCCC, play Rats (+1), draw Doctor, trash Doctor, buy Rats (+1)
T4: RRRCC, play Rats (+1), draw Copper, trash Copper, play Rats (+1), draw Copper, trash Copper, play Rats (+1), draw Rats, trash Copper, Play Rats(+1), etc.