Preheat a grill for direct, medium-high grilling. If using charcoal, push 2/3 of hot coals to 1 side; spread remaining 1/3 on other side. If using a gas grill, reduce heat on 1 side to medium.
In a large nonstick skillet over medium heat, warm 3 tsp. olive oil. Sauté garlic until fragrant, about 30 seconds. Add spinach and sauté until spinach wilts, 1 to 2 minutes. Transfer to a small bowl.
Dust a 14-inch-wide pizza peel with flour or cornmeal; flour a work surface. Roll dough into a 12-inch round. Transfer dough to pizza peel; brush with remaining 2 tsp. oil.
Have toppings within reach of grill. Slide dough onto hottest part of grill; cover grill. Cook, rotating, until bottom is blistered and browned, about 2 minutes. Flip dough onto cooler side of grill, cooked side facing up. Spread pizza sauce on crust; top with mozzarella, leaving a 1/2-inch border. Spread spinach mixture on top; sprinkle with feta.
Cover grill and cook until bottom crust has browned and cheese has melted, 2 to 3 minutes, occasionally uncovering and checking bottom of crust to ensure it's not burning. If it is, reduce heat (if using a gas grill) or move off the grill (if using charcoal). Transfer pizza to a cutting board. Let stand for 2 minutes, then slice and serve.