In a saucepan over medium-high heat, heat 1 tablespoon butter until foamy, then add the onion and season with 1/8 teaspoon of salt and a few grinds of pepper. Cook, stirring often, until the onion is softened, 4-5 minutes.
Transfer the onion to a heatproof mixing bowl. Whisk in the eggs, milk, remaining 1/8 teaspoon of salt, and pepper.