Boost Shake says it can be used during my first turn or on a Pokemon that was put into play this turn. What if my Basic Pokemon was in play last turn but I evolve it into a Stage 1 this turn; can I use Boost Shake on that Pokemon to search for a Stage 2 and put it onto that Pokemon?
Yes, you can.
Source: TPCi Rules Team (2021-09-09)

Relevant cards

Note: All applicable cards may not be listed below.
SWSH7_142
SWSH7_142