Melt 1 tablespoon butter in an 8-inch nonstick skillet over medium heat. Season eggs with salt and pepper.
Add half of eggs to skillet. Cook eggs, stirring gently with a heatproof silicone spatula, until eggs are lightly scrambled and almost cooked, about 3 minutes. Spread eggs evenly to cover bottom of skillet.
Top eggs with half of ricotta, Parmesan, basil, and chives. Using spatula, fold up one-third of omelet. Roll omelet over onto itself, then slide omelet onto a plate. Repeat with remaining ingredients to make a second omelet. Top with Cherry Tomato Vinaigrette.
