Step 1

Preheat oven to 375°F. Pour tomatoes with juices into a 13- x 9-inch baking dish. Stir in 1/4 teaspoon salt and 1/4 teaspoon pepper. Swirl 1/4 cup water into empty can to capture juices; set aside.

Step 2

Cut tops off garlic heads to expose cloves; sprinkle with remaining 1/8 teaspoon salt and remaining 1/8 teaspoon pepper. Nestle garlic, cut side up, in center of tomatoes. Drizzle garlic and tomatoes with oil, making sure exposed cloves are coated in oil. Cover baking dish with aluminum foil. Roast in preheated oven 45 minutes. Remove from oven, and carefully remove foil. (Steam will be released.) Increase oven temperature to 425°F, and immediately return baking dish to oven. Bake at 425°F until garlic is soft and caramelized in spots, 20 to 25 minutes. Remove from oven, and let cool slightly, about 15 minutes.

Step 3

Squeeze garlic cloves from skins into tomato mixture; discard skins. Transfer tomato-garlic mixture to a blender. Process until smooth, about 15 seconds. (Alternatively, mash tomatoes and garlic together in a medium bowl until well combined.) Transfer 1 1/2 cups sauce to a large skillet or saucepan; freeze remaining sauce for another use. Stir butter and reserved tomato water in can into sauce in skillet. Cover and cook over medium, undisturbed, 5 minutes. Remove from heat.

Step 4

Pulse flour, 3/4 cup warm water, and 1 tablespoon oil in a food processor until ingredients are evenly distributed and couscous-like beads of dough form, about 5 pulses. Mixture should come together easily when pressed; if too dry, add warm water, 1 teaspoon at a time, pulsing after each addition. If too sticky, pulse in a dust- ing of flour. Transfer dough to a work surface. Knead until smooth and firm, 5 to 10 minutes. Wrap tightly in plastic wrap. Let rest at room temperature 1 hour.

Step 5

Dust 2 large baking sheets with semolina flour. Cut a one-quarter portion from dough; rewrap remaining dough to prevent it from drying out. Flatten smaller dough portion using palm of your hand. Using a rolling pin or wine bottle, roll dough into a 1/4-inch-thick rectangular slab. Brush slab on all sides with 1 teaspoon oil to prevent drying and cracking. Cut slab into 1/4-inch- wide strips. Using your fingers, roll each strip into a thin, rustic strand (2 to 3 millimeters thick). Coil each strand into a spiral; place on prepared baking sheet. Repeat process with remaining wrapped dough and oil, cutting 1 portion at a time and keeping remaining dough portions covered.

Step 6

Bring a large pot of water to a boil over high; season with salt as desired. Carefully unfurl pici, shaking off any excess semolina, and drop into boiling water. Stir gently to prevent sticking. Cook, undisturbed, until tender, about 3 minutes. While pici cook, uncover sauce, and reheat over medium-low, stirring often.

Step 7

Using tongs, transfer cooked pici to sauce in skillet, reserving cooking liquid in pot. Cook mixture, gradually adding cheese and stirring constantly, until cheese is melted, about 1 minute. Loosen sauce by adding reserved cooking liquid, 1/4 cup at a time, if needed. Divide pici evenly among 4 bowls. Sprinkle with additional cheese, and drizzle with additional oil. Garnish with basil, if desired, and serve imm diately.