Position rack in the middle of the oven; preheat to 400°F. Line a large rimmed baking sheet with parchment paper or foil.
Place squash halves cut-sides up on the prepared baking sheet. Rub cut sides with 1 tablespoon oil. Roast until the squash flesh is tender, 50 minutes to 1 hour. (Alternatively, place squash cut-side down in a microwave-safe dish; add 2 tablespoons water. Microwave, uncovered, on High until the flesh is tender, 10to 12 minutes.)
Meanwhile, heat the remaining 3 tablespoons oil in a large skillet over medium heat. Add sliced garlic and ¼teaspoon crushed red pepper; cook, stirring often, until the garlic is just starting to brown around the edges, 2 to 3 minutes. Add sliced basil and 2 cups sliced tomatoes: cook, stirring often, just until the tomatoes begin to soften, about 2 minutes. Remove from heat and stir in rinsed beans and ¾ teaspoon salt.
Using a fork and tongs, scrape the long strands of squash flesh into the skillet with the tomatoes; toss well to combine. Return the mixture to the squash shells, then top with the remaining ¼ cup sliced tomatoes and the sliced mozzarella. Bake until the cheese melts, about 10 minutes. Cut each squash half in half width-wise. Drizzle with 1 tablespoon balsamic glaze