Use Frozen Butter for Omelets
Most of us were taught to make scrambled eggs with milk. The science behind this is that the lipids in the milk coat the protein in the egg and help keep all the moisture from cooking out, resulting in a better texture. For omelets though, which are a more homogenized single piece, milk can add too much moisture. Adding small (shredded, perhaps) pieces of frozen butter to your egg before cooking will solve this problem, and the butter will melt evenly while cooking.