Heat oven to 375°F. Fit dough into a 9-inch pie dish, crimp edges, and chill while you start the onions.
Warm butter and oil in a wide skillet over medium-low heat.
Add onions, half the salt, and sugar. Cook slowly, stirring often, until deep golden, 25–30 minutes.
Deglaze with balsamic, scraping up browned bits. Let onions cool slightly off heat.
Blind-bake the crust: line with parchment, add pie weights, and bake 12 minutes. Remove weights and bake 5 minutes more.
Whisk eggs, cream, milk, Dijon, thyme, nutmeg, pepper, and remaining salt until smooth.
Scatter half the cheese over the warm crust. Spread onions evenly, then pour in the custard. Top with remaining cheese.
Bake until the center is just set with a slight wobble, 30–35 minutes. Tent edges if browning fast.
Cool 15 minutes before slicing. Serve warm or at room temperature.