Preheat oven to broil with rack 5 inches from heat. Toss together zucchini and 3 tablespoons oil in a bowl. Arrange zucchini coins in a single layer on 2 large rimmed baking sheets. Broil zucchini, 1 pan at a time, until tender and browned, 5 to 10 minutes, flipping halfway through cooking. Sprinkle with ¼ teaspoon salt. Set aside.

Bring a large pot of salted water to a boil over high. Add spaghetti; cook 1 minute less than package instructions for al dente, about 10 minutes. Drain, reserving 1 ½ cups pasta cooking liquid; set aside.

Meanwhile, heat remaining 1 tablespoon oil in a large skillet over medium. Reserve ¼ cup broiled zucchini coins for garnish. Add remaining zucchini coins to skillet; cook, stirring occasionally, until zucchini is softened and some pieces begin to break apart, about 2 minutes. Add butter, garlic, and remaining ¾ teaspoon salt; cook, stirring often, until garlic is light golden, about 3 minutes.

Add ⅓ cup reserved pasta cooking liquid to skillet; cook over medium, stirring constantly, until liquid is mostly evaporated, about 1 minute. Add cooked spaghetti and ⅓ cup pasta cooking liquid; stir, turning pasta constantly until well coated with zucchini mixture. Remove from heat; add caciocavallo, Parmigiano-Reggiano, sliced basil, and ⅓ cup pasta cooking liquid. Stir constantly until sauce thickens and pasta is evenly coated. (Add additional pasta cooking liquid if needed, ¼ cup at a time, to thin sauce to desired consistency.)

Garnish pasta with reserved zucchini coins, additional basil leaves, and grated Parmigiano-Reggiano. Serve immediately.