A baking term.

Cutting in, in baking terms, refers to blending a solid fat into dry ingredients. The fat might be butter, vegetable shortening, or lard. Cutting in is typically done by a home cook with a hand-held pastry blender, a fork, or your scrupulously clean and well-floured fingers.

The idea, in cutting in, is to break the fat down into very small particles and coat these particles with the dry ingredients. In the final product, this contributes to a smooth, rich, even texture. If the fat is not cut in well enough, you may wind up with large pockets of melted fat which are not only disagreeable in taste and texture, but may also detract from the structural soundness of whatever you are trying to bake.

A dancing term.

Typically, the phrase "May I cut in?" is said by a male who has approached a dancing couple. If the answer is yes, the first male will step away, and the new male will continue the dance with the female.

This question can sometimes be asked aggressively. Most frequently, though, it is heard at weddings and said in a playful fashion, since everyone wants to have a turn dancing with the bride.