Preheat oven to 375°, grease a large rimmed baking sheet.
Combine zucchini, garlic and mushrooms to food processor and pulse until uniform and finely chopped, about 45 seconds; transfer to a medium bowl.
Heat 1 tablespoon oil in a large nonstick skillet over medium. Add zucchini mixture and 1 1/2 teaspoon salt and cook, spreading mixture flat and stirring occasionally, until beginning to brown and liquid evaporates, about 12 minutes. Transfer to a large bowl. Add egg, breadcrumbs and basil.
Form the zucchini mixture into 16 balls about 1 ounce each, rolling tightly. Transfer them to the prepared baking sheet. Bake until firm and browned, about 20 to 25 minutes.
Serve with quick marinara sauce and pasta, if desired.