Heat butter and 1 tablespoons of olive oil a large dutch oven over medium high heat. Add in the sliced onions and season with salt and pepper. Cook, stirring occasionally, for 15 mins uncovered. Reduce heat to medium and cook another 10 minutes stirring frequently. Drop heat to medium low and continue to cook till onions are deep golden brown another 5-10 minutes. Stir in thyme leaves and remove from heat. Place in a bowl to cool.
Preheat oven to 400 degrees F. Adjust oven racks to upper and lower thirds of the oven. Line two baking sheets with parchment paper.
Combine both cheeses and diced prosciutto in a bowl.
Remove the thawed puff pastry from the refrigerator and on a lightly flowered surface roll each piece out to a 9x14 rectangle
Place each piece of puff pastry on its own baking sheet and poke all over with a fork. Cover with another piece of parchment , followed by a piece of foil and place another baking sheet on top of the foil covered puff pastry to prevent it from getting too puffy.
Bake in the oven for 15 minutes; remove the top sheet pan along with the parchment and foil. Continue to cook another 5-8 minutes until it’s golden brown, you can flip it over at the point if you notice the bottom is darker. Remove from the oven.
Drizzle each flatbread with olive oil and spread with 1/2 of the onion mixture, sprinkle each with 1/2 of the cheese and prosciutto mixture, put back into the oven and cook 5 minutes until cheese is melted and starts to brown in spots.
Remove from oven, sprinkle with fresh nutmeg, salt and pepper. Cut into desired pieces and serve hot from the oven.