In a large skillet, melt the butter over medium-high heat.
Add onions and thyme sprigs; season with 1 teaspoon salt and ½ teaspoon pepper and cook, stirring occasionally, until the onions are translucent and lightly golden on the edges; 20 to 25 minutes.
Reduce the heat to medium, add broth and sherry and cook until the onions are browned and the liquid has mostly evaporated but the mixture is still saucy; 16 to 18 minutes.
Transfer to a bowl; set aside to cool for at least 30 minutes.
Pre-heat the oven to 375°F.
Remove the dough from the refrigerator and, using a sheet of parchment paper, roll into a 13-inch round.
Spread the cooled caramelized onions on the dough, leaving a 1-inch to 2-inch border.
Fold the edges in, over the onions, transfer to a baking sheet and bake until the dough is golden brown and some of the onions have browned on the edges; 40 to 50 minutes; rotating the galette halfway into the baking process.
Remove galette from the oven and sprinkle remaining ½ cup grated Gruyère on the crust.
Place the galette back in the oven and bake another 5 minutes to melt the cheese.
Remove and let rest for 10 minutes before slicing.
Top with remaining thyme leaves, for garnish.