Start the caramelized onions first since they take time. Melt the butter with the olive oil in a large skillet over medium heat. Add the sliced onions, salt, and sugar and stir to coat. Cook for 30 to 35 minutes, stirring every few minutes, until the onions are deeply golden brown and jammy.
Add the white wine and thyme leaves during the last 2 minutes and stir until the wine has evaporated. Set aside.
While the onions cook, make the dough. Whisk together the gluten free flour, instant yeast, sugar, salt, and xanthan gum in a large bowl. Add the warm water, olive oil, and apple cider vinegar and stir until a soft dough forms. Cover and let it rise for 30 minutes.
Preheat your oven to 475 F. Line a large baking sheet with parchment paper. Transfer the dough onto the parchment and press it into a 13 to 14-inch round using oiled hands, creating a slightly thicker edge for the crust.
Brush the surface of the dough lightly with olive oil. Scatter the shredded mozzarella evenly over the crust, leaving a small border. This creates a melty base layer that keeps the crust from getting soggy.
Spread the caramelized onions evenly over the mozzarella layer. Top generously with the shredded gruyere and grated parmesan, making sure the onions are mostly covered with cheese.
Bake for 13 to 15 minutes until the crust is golden and crisp, the gruyere is melted and bubbly, and there are gorgeous golden-brown spots on the cheese.
Remove from the oven and finish with fresh thyme leaves and a sprinkle of flaky sea salt. Let it rest for 2 minutes, then slice and serve while the cheese is still stretchy and molten.