Instructions

for the dough

Process whole-wheat flour, bread flour, honey, and yeast in food processor until combined, about 2 seconds. With processor running, add water and process until dough is just combined and no dry flour remains, about 10 seconds. Let dough stand for 10 minutes.
Add oil and salt to dough and process until it forms satiny, sticky ball that clears sides of workbowl, 45 to 60 seconds. Remove from bowl and knead on oiled countertop until smooth, about 1 minute. Shape dough into tight ball and place in large, lightly oiled bowl. Cover tightly with plastic wrap and refrigerate for at least 18 hours or up to 2 days.

for the cream sauce

Heat oil in 10-inch skillet over medium heat until shimmering. Add shallots and salt and cook until softened, about 5 minutes. Add garlic and thyme and cook until fragrant, about 30 seconds. Stir in cream and bring to simmer. Cook until cream is thickened and spatula leaves trail that slowly fills in, about 3 minutes. Remove skillet from heat. Discard thyme sprig and stir in Parmesan and pepper. Transfer to bowl and let cool completely before using. (Cream sauce can be refrigerated, covered, for up to 2 days.)

for the mushrooms

Heat oil in 12-inch nonstick skillet over medium-high heat until shimmering. Add mushrooms, salt, and pepper and cook, stirring frequently, until liquid evaporates and mushrooms are lightly browned, 5 to 7 minutes. Transfer to bowl and let cool completely before using. (Mushroom mixture can be refrigerated, covered, for up to 2 days.)
One hour before baking pizza, adjust oven rack 4½ inches from broiler element, set pizza stone on rack, and heat oven to 500 degrees. Remove dough from refrigerator and divide in half. Shape each half into smooth, tight ball. Place balls on lightly oiled baking sheet, spacing them at least 3 inches apart. Cover loosely with plastic coated with vegetable oil spray; let stand for 1 hour.
Heat broiler for 10 minutes. Meanwhile, coat 1 ball of dough generously with flour and place on well-floured countertop. Using your fingertips, gently flatten into 8-inch disk, leaving 1 inch of outer edge slightly thicker than center. Lift edge of dough and, using back of your hands and knuckles, gently stretch disk into 12-inch round, working along edges and giving disk quarter turns as you stretch. Transfer dough to well-floured peel and stretch into 13-inch round. Using back of spoon, spread half of cream sauce over surface of dough, leaving ¼-inch border. Sprinkle half of mushroom mixture evenly over pizza, followed by 3/4 cup fontina. Slide pizza carefully onto stone and return oven to 500 degrees. Bake until crust is well browned and cheese is bubbly and partially browned, 8 to 10 minutes, rotating pizza halfway through baking. Remove pizza, place on wire rack, and let rest for 5 minutes. Top with half of arugula, slice, and serve with lemon wedges.
Heat broiler for 10 minutes. Repeat process of stretching, topping, and baking with remaining dough and toppings, returning oven to 500 degrees when pizza is placed on stone.