Covering Problem

As promised, the follow up problem:

Find a connected arrangement of squares such that it is not possible to tile it with 2x1 dominoes, but if you increase the shape by adding a 2x1 shape to it, you can now tile the new shape with 2x1 dominoes.

Naturally, I mean 'tile' in the sense of the domino problem. Note that the solution to this problem requires a connected shape. I hope I don't need to define connected, I mean, I could, but its annoying.