In a large bowl combine yeast and water. Let stand until yeast is dissolved; about 5 minutes.
Add 2 cups all-purpose flour, semolina flour, 3 tablespoons olive oil, salt and sugar. Mix by hand or on low speed of an electric mixer until well-combined. Turn out onto a lightly floured surface. Knead in enough of the remaining all-purpose flour for 6 to 8 minutes or until a smooth elastic dough is formed.
Transfer dough to a bowl lightly coated with additional olive oil, turning to coat all sides. Cover and let rise in a warm place until about doubled in size, 1 to 1 1/2 hours.
Meanwhile, in a large skillet over medium-high cook and stir mushrooms in hot butter for 2 to 3 minutes until mushrooms just begin to soften. Add shallots, wine and thyme. Cook and stir for 4 to 6 minutes or until shallots are tender. Add asparagus; cook and stir for 2 minutes. Remove from heat.
Preheat oven to 475°F. Grease and dust 2 extra-large baking sheets with cornmeal. Punch dough down and divide dough in half. On a lightly floured surface roll out one portion of the dough to 1/8-inch thickness (don't worry if its not perfectly round). Transfer to one prepared baking sheet.
Brush lightly with additional olive oil. Top with half the mushroom mixture and half the cheese; season to taste with pepper. Bake in oven on lowest rack about 12 minutes until crispy and brown. While first pizza is baking, repeat with remaining dough and toppings.
I have been making this for the past two years! My family and I love this pizza. I needed something unique to do with my spring finds and this hits the spot!