French Onion Rolls

Ingredients

• 3/4 cup (180g) whole milk
• 1/2 cup (120g) water
• 1/3 cup (75g) unsalted butter, cubed
• 4 to 4 1/2 cups (500 to 531g) all-purpose flour, divided, plus more for dusting
• 2 tablespoons (24g) granulated sugar
• 1 tablespoon (9g) kosher salt (see note)
• 1 (0.25oz) package (7g) or 2 1/4 teaspoons Red Star Active Dry Yeast
• 1 large (50g) egg, room temperature
• 1/3 cup (75g) unsalted butter, softened
• 2 tablespoons (4g) fresh thyme leaves
• 1/2 cup (56g) shredded Gruyère cheese
• Caramelized Onions (see below)
• 2 tablespoons (28g) unsalted butter
• 1 pound (454g) sweet yellow onions, thinly sliced (about 9 cups)
• 1/2 teaspoon (1.5g) kosher salt (see note)
• 1/2 teaspoon (2g) granulated sugar
• 1/2 teaspoon (1g) ground black pepper
• 1/4 cup (60g) dry white wine
• 1 tablespoon (15g) water