Heat oil in 8-inch skillet over medium-low heat until shimmering. Add garlic, pepper, thyme, and salt and cook, stirring constantly, until fragrant, about 30 seconds. Transfer to bowl and let cool completely before using.
Combine figs and water in bowl. Cover and microwave until steaming, 1 to 2 minutes. Let figs stand, covered, until softened and plump, about 5 minutes. Strain figs; discard water
One hour before baking, adjust oven rack 4 inches from broiler element, set baking stone on rack, and heat oven to 500 degrees. Press down on dough to deflate. Transfer dough to clean counter, divide in half, and cover loosely with greased plastic. Pat 1 piece of dough (keep remaining piece covered) into 4-inch round. Working around circumference of dough, fold edges toward center until ball forms. Flip dough ball seam side down and, using your cupped hands, drag in small circles on counter until dough feels taut and round and all seams are secured on underside. (If dough sticks to your hands, lightly dust top of dough with flour.) Repeat with remaining piece of dough. Space dough balls 3 inches apart, cover loosely with greased plastic, and let rest for 1 hour.
Heat broiler for 10 minutes. Meanwhile, generously coat 1 dough ball with flour and place on well-floured counter. Using your fingertips, gently flatten into 8-inch round, leaving 1 inch of outer edge slightly thicker than center. Using your hands, gently stretch dough into 12-inch round, working along edge and giving disk quarter turns.
Transfer dough to well-floured pizza peel and stretch into 13-inch round. Using back of spoon, spread half of garlic oil in even layer on surface of dough, leaving 1/4-inch border around edge. Layer 1/2 cup basil leaves over garlic oil. Sprinkle with 1/2 cup feta, followed by half of the figs.
Slide pizza carefully onto baking stone and return oven to 500 degrees. Bake until crust is well browned and cheese is partially browned, 8 to 10 minutes, rotating pizza halfway through baking. Transfer pizza to wire rack and drizzle 1 tablespoon honey over surface. Let cool for 5 minutes before slicing and serving. Heat broiler for 10 minutes. Repeat with remaining dough, garlic oil, and toppings, returning oven to 500 degrees when pizza is placed on stone.