Step 1: In a large skillet over medium heat, melt butter with olive oil. Add sliced onions and cook for 10 minutes, stirring occasionally.
Step 2: Sprinkle sugar, salt, and pepper over onions. Reduce heat to low and cook for 20-25 minutes, stirring occasionally, until onions are deeply golden and caramelized. Remove from heat and let cool.
Step 3: Preheat oven to 375°F (190°C). Line a baking sheet with parchment paper.
Step 4: On a lightly floured surface, roll pizza dough into a 12×16-inch rectangle.
Step 5: Spread cooled caramelized onions evenly over dough, leaving a 1-inch border around edges. Sprinkle shredded Gruyere over onions.
Step 6: Starting from the long side, tightly roll dough into a log. Pinch seam to seal.
Step 7: Transfer log to prepared baking sheet and shape into a circle, joining ends to form a wreath. Pinch ends together to seal.
Step 8: Using kitchen shears, make cuts about 1 inch apart around the wreath, cutting 3/4 of the way through dough. Gently twist each section to expose filling.
Step 9: Brush wreath with beaten egg wash. Bake for 25-30 minutes until golden brown and cheese is bubbly.
Step 10: Let cool for 10 minutes, garnish with thyme if desired, then serve warm.