In a large mixer, combine flour, yeast, salt, and water. Mix until fully combined, then let rest, covered, for ten minutes. Turn mixer back on for ten more minutes to develop the gluten. Remove to an oiled bowl, cover, and let rest for an hour or until doubled in size.
Preheat oven to 500 degrees F.
Press dough into a square 9x9-inch pan, using fingers to push out any air bubbles. Top with cheese cubes and mushrooms, making sure cheese goes all the way to the edges (to melt down the sides and form the crunchy crust. Add sauce on top of cheese and mushrooms and bake for 5-10 minutes (or until cheese is browning).
Remove from pan and use metal spatula to loosen edges and remove from pan. Cut into four slices (two each).