Whisk the eggs with 2 tablespoons water in a large bowl until frothy. Add the butter to a large nonstick pan over medium-low heat. (If it immediately sizzles, the pan is too hot. It should be a slow melt.) Tilt the pan to swirl the butter and coat the bottom and sides of the pan.
Add the whisked eggs to the pan and allow to rest until a light layer of cooked egg develops on the bottom, 2 to 3 minutes. Use a whisk to stir and scramble the eggs, then let rest again. Repeat this until the eggs are about three-quarters of the way done, then add the salsa. Raise the heat slightly and cook, whisking the salsa into the eggs. Cook to desired texture. Top with cheese and cilantro.