In a large mixing bowl combine yeast and the warm water. Let stand about 5 minutes or until yeast is dissolved.
Add 2 cups of the all-purpose flour, the semolina flour, the 3 tablespoons oil, the sugar, and salt. Beat with an electric mixer on low speed or by hand until combined, scraping sides of bowl constantly. Turn dough out onto a lightly floured surface. Knead in enough of the remaining all-purpose flour to make a moderately stiff dough that is smooth and elastic (6 to 8 minutes total). Shape dough into a ball. Place in a lightly greased bowl, turning once to grease surface of dough. Cover and let rise in a warm place until double in size (1 to 1 1/2 hours).
Meanwhile, in a large skillet heat butter over medium-high heat until melted. Add mushrooms; cook for 2 to 3 minutes or just until mushrooms begin to soften, stirring occasionally. 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 more. Remove from heat.
Punch dough down. Turn dough out onto a lightly floured surface; divide in half. Roll each dough portion into an 1/8-inch-thick circle (don't worry if it's not perfectly round). Brush bottom of crusts lightly with additional oil. Invert two large baking sheets; brush the back of baking sheets with additional oil and dust with cornmeal. Place crusts on the prepared baking sheets.
For a charcoal or gas grill, place one of the crusts on the rack of a covered grill directly over medium-low heat. Grill for 1 to 2 minutes or until dough is puffed in some places and starting to become firm. Using tongs, carefully turn crust over and transfer to the back of a baking sheet. Quickly brush grilled side of crust with additional oil.
Top crust with half of the mushroom mixture and half of the cheese; sprinkle with pepper. Transfer pizza from baking sheet to grill rack. Grill for 3 to 5 minutes or until crust is crisp and cheese is melted. Repeat with the remaining crust and toppings.