Make dough: In a medium saucepan, heat milk and ½ cup (120 grams) water over medium heat to scalding. (An instant-read thermometer should register 180°F/82°C.) Remove from heat and add cubed butter; stir occasionally until butter is melted, and an instant-read thermometer registers 120°F (49°C) to 130°F (54°C).
In the bowl of a stand mixer fitted with the paddle attachment, beat 2 cups (250 grams) flour, sugar, salt, and yeast at low speed until combined. Add warm milk mixture and beat until combined. Add 1 egg (50 grams), beating until combined. With mixer on low speed, gradually add 2 cups (250 grams) flour, beating just until a shaggy dough comes together, stopping to scrape sides of bowl.
Switch to the dough hook attachment. Beat at low speed until a soft, somewhat sticky dough forms, about 24 minutes, stopping to scrape bottom and sides of bowl and dough hook; add up to remaining ¼ cup (31 grams) flour, 1 tablespoon (8 grams) at a time, if dough is too sticky. Turn out dough onto a clean surface; knead 5 to 6 times, and shape into a smooth round.
First rise: Lightly oil a large bowl. Place dough in bowl, turning to grease top. Cover and let rise in a warm, draft-free place (75°F/24°C) until doubled in size, about 1 hour.
Make Caramelized Onions: In a Dutch oven or large saucepan, melt butter over medium heat. Add onion; cook, stirring occasionally, until softened, about 15 minutes. Stir in salt, sugar, and pepper; cook, stirring occasionally, until onion is deep golden brown, 30 to 40 minutes. Increase heat to high. Add wine; cook, stirring frequently, until most of liquid has evaporated. Transfer to a heatproof bowl and let cool completely.
Make filling (butter mixture): In a small bowl, stir together softened butter and thyme. Set aside.
Prepare pan: Lightly spray a 12-cup muffin pan with baking spray with flour.
Make rolls: Punch down dough and turn out onto a lightly floured surface. Roll dough into a 12x18-inch rectangle, one long side facing you.
Using a small offset spatula, spread butter mixture onto dough, leaving a ½-inch border on long side facing you. Spread Caramelized Onions evenly onto butter mixture; sprinkle with ½ cup (57 grams) cheese. Fold top half down, lightly pressing to adhere. Trim short sides; cut into 12 equal strips (about 1½ inches each).
Cut 1 strip into 3 equal strands, leaving ½ inch from the folded end intact. Braid strands loosely, pinching end to adhere. Carefully roll ends of braid under to create a knotted roll. Place in a prepared muffin cup. Repeated with remaining dough strips.
Second rise: Cover and let rise in a warm, draft-free place (75°F/24°C) until puffed, 30 to 45 minutes.
Preheat oven to 375°F (190°C).
Make Egg Wash: In another small bowl, whisk together water and egg. Using a pastry brush, carefully brush onto dough.
Bake rolls: Bake for 10 minutes. Sprinkle remaining ½ cup (57 grams) cheese onto rolls and bake until golden brown and an instant-read thermometer inserted in center registers at least 190°F (88°C), 8 to 10 minutes more. Let cool in pan for 10 minutes. Serve warm, or remove from pan, and let cool to room temperature on a wire rack. Refrigerate in an airtight container for up to 3 days.