Pat the tilapia dry with paper towels, then season with 1/2 teaspoon salt and a few grinds of pepper. Spread the flour on a plate; dredge the tilapia in the flour, tapping off the excess. Heat 2 tablespoons olive oil in a large nonstick skillet over medium-high heat. Add the tilapia and cook until golden brown, about 3 minutes per side. Transfer to a plate; tent with foil to keep warm. Wipe out the skillet.
Heat 3 tablespoons olive oil in the skillet over medium heat. Add the garlic and cook, stirring, until it starts browning, about 2 minutes. Add the chicken broth, wine, and lemon zest and juice. Increase the heat to high, bring to a boil and cook until the liquid is reduced by half, about 5 minutes; season with salt and pepper. Add the butter and whisk until slightly thickened, about 1 minute; stir in the parsley.
Meanwhile, toss the spinach with the remaining 1 tablespoon olive oil and a few grinds of pepper. Divide among plates, top with the fish and drizzle with some pan sauce. Serve with lemon wedges.