Heat an air fryer (at least 5-quart capacity if using a basket-style fryer) to 400°F. Cut 1 small spaghetti squash in half lengthwise and scoop out the seeds. Brush the cut sides of the squash with 1 teaspoon of the olive oil and season with 1/2 teaspoon of the kosher salt and several grinds black pepper.
Arrange the squash cut side up in the air fryer. Cook until the tops of the squash are beginning to brown, the shells have shriveled slightly, and a paring knife slides easily through the flesh, 20 to 25 minutes. Meanwhile, prepare the toppings.
Shred 3 ounces mozzarella cheese on the large holes of a box grater (about 3/4 cup) and grate 1/2 ounce Parmesan cheese on the smallest holes (1/4 cup). Pick the leaves from 2 to 3 small fresh basil sprigs and finely chop until you have 2 tablespoons fresh basil leaves, plus chop a little extra for garnish.
Place half the Parmesan in a small bowl. Add 2 tablespoons panko breadcrumbs, the remaining 1/2 teaspoon olive oil, and a pinch of kosher salt and stir to combine.
Using tongs, transfer the squash to a cutting board. When cool enough to handle, use a fork to shred the inside of each squash half, leaving the strands in the shell.
Divide the following between the squash halves: 1/2 cup marinara sauce, mozzarella, remaining Parmesan, basil, the remaining 1/4 teaspoon kosher salt, a pinch of red pepper flakes, and a few grinds black pepper. Stir everything into the strands to combine, then sprinkle both halves with the panko mixture.
Return the squash to the air fryer and cook at 400°F until the cheese is melted and the breadcrumbs are browned in spots, 5 to 7 minutes. Garnish with the reserved basil and serve hot.