Pat the tilapia dry with paper towels, then season with salt and pepper. Spread the flour on a plate; dredge the tilapia in the flour, tapping off the excess. Heat some of the 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 half of the 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 olive oil and some pepper. Divide among plates, top with the fish and drizzle with some pan sauce. Serve with lemon wedges.