Preheat oven to 400°. In a large skillet over medium-high heat, heat oil and 1 tablespoon butter just until butter begins to brown. Add chickpeas and cook, stirring occasionally, until golden and crispy, 5 to 8 minutes; season with salt and pepper. Transfer to a paper towel-lined plate; set aside.
Reduce heat to medium-low. In same skillet over medium heat, melt remaining 2 tablespoons butter. Add garlic and cook, stirring, until fragrant and very lightly golden, 1 to 2 minutes. Add flour and cook, stirring constantly, until mixture bubbles up and appears lightly golden, about 1 minute more.
Add cream and bring mixture to a low simmer. Add spinach and cook, stirring constantly, until wilted, about 5 minutes. Stir in Parmesan until melted; season with salt and pepper. Transfer about ½ cup spinach mixture to a small bowl.
Divide remaining spinach mixture between both flatbreads. Top with mozzarella.
Bake flatbreads until cheese is bubbly, about 15 minutes. Drizzle with reserved sauce and sprinkle with reserved chickpeas.
