Heat oven to 400°F. Brush large cookie sheet with oil. Unroll dough; place on oiled cookie sheet. Starting at center, press out dough into 15x10-inch rectangle. Bake 8 to 10 minutes or until crust is very light brown.
Meanwhile, in 12-inch skillet, heat butter over medium-high heat until melted. Add mushrooms; cook about 6 minutes, stirring frequently, until well browned; drain. Stir in thyme, salt and pepper.
Spread cream cheese evenly over crust. Sprinkle fontina cheese over cream cheese. Spread cooked mushrooms over cheese. Sprinkle mozzarella cheese over mushrooms.
Bake 10 to 12 minutes longer or until cheese is melted and crust is golden brown. Cool 5 minutes. Cut into 8 rows by 6 rows. Sprinkle with thyme leaves.