Place the unsalted butter and 1 tablespoon olive oil in a large Dutch oven and warm over medium-high heat. Add the sliced yellow onions, then season them with kosher salt and freshly cracked black pepper.
Cook the onions uncovered for 15 minutes, stirring from time to time. Reduce the heat to medium and continue cooking for another 10 minutes, stirring more frequently.
Turn the heat down to medium-low and cook for an additional 5–10 minutes, or until the onions develop a deep golden-brown color. Fold in the fresh thyme leaves, remove the pot from the heat, and transfer the onions to a bowl to cool.
Preheat the oven to 400°F and arrange the oven racks in the upper and lower thirds. Line two baking sheets with parchment paper.
On a lightly floured work surface, roll each thawed puff pastry sheet into a rectangle measuring approximately 9 × 14 inches. Transfer each pastry rectangle to a separate prepared baking sheet, then prick the entire surface repeatedly with a fork to limit excessive rising.
Lay another piece of parchment paper over each pastry, cover with foil, and place an additional baking sheet on top to weigh the pastry down while it bakes.
Bake the weighted puff pastry sheets for 15 minutes. Carefully remove the upper baking sheets along with the foil and top layers of parchment.
Return the uncovered pastries to the oven and bake for another 5–8 minutes, until evenly golden. If the undersides become noticeably darker than the tops, turn the pastry sheets over to encourage more uniform browning.
Remove the baked pastry from the oven and allow it to cool slightly.
While the pastry is baking, place the shredded Gruyère and Fontina cheeses in a bowl. Tear or dice the prosciutto into smaller pieces and add it to the cheeses. Toss everything together until the ingredients are evenly distributed.
Lightly drizzle the baked puff pastry sheets with the remaining 2 tablespoons olive oil. Divide the cooled caramelized onions evenly between the two pastries and spread them over the surfaces.
Distribute the Gruyère, Fontina, and prosciutto mixture equally over the onion layer. Return both flatbreads to the oven and bake for 5 minutes, or until the cheeses have melted and begun to develop browned spots.
Add additional freshly cracked black pepper at this stage if desired.
Remove the finished flatbreads from the oven. Grate fresh nutmeg lightly over the tops, then season with additional kosher salt and freshly cracked black pepper according to taste.
Slice the flatbreads into serving pieces and serve while hot.