If a Basic Pokemon was put into play last turn, can I use Evosoda on it twice to bring it all the way to a Stage 2?
No, you cannot. When you evolve a Pokemon it is a new Pokemon, and since it was put into play this turn you cannot use Evosoda on it until next turn.
Source: XY FAQ (2014-02-06), TPCi Rules Team (2014-02-06)

Relevant cards

Note: All applicable cards may not be listed below.
XY1_116