Woodsy mushrooms and molten cheese put this hearty flatbread in the red-wine camp. Open our Oakville Cabernet Sauvignon, a classic, concentrated red wine that responds to the flatbread’s earthy mushroom flavor, its cheesy richness, and the scent of thyme.
In the bowl of a stand mixer, proof the yeast in ¼ cup of the warm water until it foams. Add the remaining water, flour, and salt. With the dough hook, knead the dough for 8 minutes on medium speed.
Portion the dough into 3 equal balls. Place in a lightly oiled 9-by-13-inch pan, cover with plastic wrap, and refrigerate overnight.
Remove the dough from the refrigerator. Arrange racks on the lower third and middle of the oven and preheat to 425°F. Line 3 half sheet pans with parchment paper.
With your hands, stretch each ball of dough into a 12-by-7-inch rectangle and place on a prepared pan.
To assemble the first flatbread, spread about one-third of the crème fraîche on the dough, leaving roughly a ½-inch border. Continue with a third of the fontina and the parmesan. Scatter a third of the mushrooms on the cheese and finish with a third of the thyme.
Place the pan on the lower rack of the oven and bake the flatbread for 8 minutes. Rotate the pan and move to the middle rack. Bake for another 5 minutes until golden brown. Remove the flatbread from the pan and let cool for a few minutes. Cut into triangles, and arrange on a platter.
While the first flatbread is baking, assemble the next and so on.