Lightly spray a 12-cup muffin pan with baking spray with flour.
Punch down Do-It-All Dough, and turn out onto a lightly floured surface. Roll dough into a 18x12-inch rectangle with one long side closest to you.
In a small bowl, stir together butter and thyme. Using a small offset spatula, spread onto dough, leaving a 1⁄2-inch border on long side closest to you. Spread Caramelized Onions evenly onto butter mixture; sprinkle with 1⁄2 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 1⁄2 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. 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).
In another small bowl, whisk together egg and 1 tablespoon (15 grams) water. Using a pastry brush, carefully brush onto dough.
Bake for 10 minutes. Sprinkle remaining 1⁄2 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 completely on a wire rack. Refrigerate in an airtight container for up to 3 days.