For the pesto: Combine the basil, pine nuts, garlic and oil in a food processor and blend until a paste forms. Add the Parmesan, Pecorino and some salt and pepper and pulse until smooth. Keep covered in the refrigerator until ready to use.
For the flatbread: Measure 3/4 cups lukewarm water in a measuring cup, stir in the yeast, and let it develop for 10 minutes. It should look foamy when its ready.
Meanwhile, add the flour and salt to the bowl of a stand mixer fitted with a dough hook. Whisk for about a minute to create a well. Add the water and yeast in a steady stream while the mixer is running. Turn the mixer up to medium speed and knead until the dough comes away from the bowl and it looks smooth and feels elastic but is still a bit sticky, 5 to 7 minutes.
Grease a large glass bowl with 2 tablespoons of the olive oil. Add the dough, flip on both sides, and cover with a clean tea towel. Place the dough in a warm spot to rise until it has doubled in size, about 1 1/2 hours.
Drizzle the remaining tablespoon of olive oil in a large cast-iron pan, brushing the sides well with a brush. Stretch the dough directly in the pan to a 12-inch round, roughly the same size as your skillet. Push your fingers into the dough to leave little imprints. Let the dough sit for 10 minutes so it can puff up a little. Combine the pesto and yogurt in a bowl and reserve.
Begin setting up your campfire. Place a grill grate over the fire or use logs to make a nice place to put the skillet over the heat.
Place the skillet over the fire and cover with a lid. (Or alternatively, place it over a burner on the stove set to medium heat.) Cook the flatbread for 12 minutes, until the bottom is golden and slightly crisp, then flip, and cook, covered, until puffy and golden and cooked through completely, 8 to 10 minutes. Alternatively, you can also cook the flatbread in the oven. Preheat the oven to 400 degrees F. Place the skillet on the middle rack and bake until puffed and golden, 20 to 25 minutes.