Normally the way a spoon would be kashered (in most circumstances) is taking a pot of boiling water, then dropping the spoon in.
Instead, could I put a spoon in a pot of cold water, then bring the whole thing to a boil?
Apparently not. Rema (Orach Chaim 452:1) says that "hag'alah is worthless if the water isn't boiling," and that therefore "one must be careful not to insert any dishes [to be kashered] until the water boils." (Chok Yaakov there adds that, for the same reason, there must be a pause between putting in one utensil and the next, since the insertion of a cold object cools off the water for a few seconds.)
The problem, it seems, would be that if it's done the way you suggest, then the pot will absorb the non-kosher substance from the spoon, and then the spoon will re-absorb it from the pot (or from the water).