Preheat oven to 400°F (200°C).
Melt butter in a large oven-safe skillet over medium heat. Add onions, salt, and pepper. Cook, stirring occasionally, until caramelized, about 25 minutes.
Add broth and cook until mostly evaporated, about 2 minutes.
In a large bowl, whisk together eggs, milk, mustard, and 1/2 cup of the cheese.
Pour egg mixture into the skillet over the onions. Stir gently to combine.
Sprinkle the remaining cheese on top and transfer the skillet to the oven.
Bake until the frittata is puffed and golden, 15–18 minutes.
Let cool slightly, garnish with thyme if using, slice, and serve.