Sprinkle the yeast over the warm water in a large bowl and stir well. When small bubbles start to form on the surface, after 5 minutes or so, add 1 cup of the flour and stir well. Let rest for 10 minutes.
Add another cup of the flour, the oil, rosemary, and salt, and stir until you have a shaggy mass. Turn out onto a well-floured work surface and sprinkle the dough with a teaspoon or so of flour. Allow the dough to rest and absorb the flour for about 10 minutes. Wash the bowl, dry it well, and lightly oil it.
Using a spatula or a bench scraper, gently lift, fold, and press down the dough, then give it a quarter turn. Continue this gentle kneading until the dough is smooth and elastic, about twelve turns, adding as little additional flour as possible. This is a wet, sticky, dough.
Place the dough in the bowl and turn to coat in the oil, then cover with a tea towel and let rise for an hour. (Or place the covered bowl in the refrigerator for up to 2 days for a slow rise. Bring to room temperature before proceeding.)
Heat the 2 tablespoons olive oil in a large sauté pan over low heat. Add the onions and season with plenty of salt and pepper, and cook slowly until well browned and caramelized, 14 to 20 minutes. Remove from the heat.
Preheat the oven to 450° F. Line a baking sheet with parchment. Brush the parchment with oil.
Place the dough on the parchment-lined baking sheet and press out into a 14- by 10-inch rectangle. If the dough shrinks back and fights you, let it relax for 10 minutes, then press it out gently. Dimple the focaccia all over with your fingertips (as though you were lightly playing the piano). Let rest for 20 minutes.
Brush the focaccia lightly with olive oil. Spread the preserves to within 1/2 inch of the edges using an offset spatula. Cover the preserves with the caramelized onions. Break up the cheese and crumble it over the Focaccia. Drizzle with olive oil and sprinkle with crunchy salt and pepper.
Bake the focaccia until it is golden brown 20 to 25 minutes
Serve warm or at room temperature.
