Combine 1/2 cup warm water and yeast in the bowl of a stand mixer with dough hook attached; let stand 5 minutes or until bubbly. Add 4 teaspoons oil and 1/4 teaspoon salt to yeast mixture. Weigh or lightly spoon flour into dry measuring cups; level with a knife. Sprinkle flour over yeast mixture; mix 2 minutes or until a soft dough forms. Place dough in a large bowl coated with cooking spray; cover surface of dough with plastic wrap lightly coated with cooking spray. Refrigerate 24 hours.
Remove dough from refrigerator. Let stand, covered, 1 hour or until dough comes to room temperature. Punch dough down. Roll dough out to a thin 12-inch circle on a lightly floured baking sheet, without raised edges, sprinkled with cornmeal. Crimp edges to form a 1/2-inch border. Pierce dough several times with a fork. Cover dough loosely with plastic wrap.
Position an oven rack in the lowest setting. Place a pizza stone on lowest rack. Preheat oven to 550°. Preheat the pizza stone for 30 minutes before baking dough.
Combine 1 tablespoon oil, thyme, pepper, apricots, shallots, and remaining 1/4 teaspoon salt; toss gently. Remove plastic wrap from dough; slide dough onto preheated pizza stone, using a spatula as a guide. Bake at 550° for 4 minutes. Top dough with goat cheese and apricot mixture. Bake an additional 5 minutes or until crust is golden brown. Cut pizza into 10 slices; sprinkle with parsley and chives. Toss arugula with remaining 1 1/2 teaspoons oil; arrange arugula over apricot mixture. Top with prosciutto and Parmigiano-Reggiano cheese. Cut into 10 wedges.
Wine note: When a dish has an ingredient that is also a signature flavor in a wine, go with the match. For this pizza, it's viognier, a full-bodied, aromatic white that often announces itself with apricot aromas. Try the Bridlewood 2008 Reserve Viognier from California's central coast ($24). Under its stone fruit flavors lurk layers of citrus—tangerine and tangy grapefruit—with an elegant honeysuckle wrap. Rich enough to love the prosciutto on the pizza, the wine has lovely acidity that works well with goat cheese, too. —Sara Schneider