Position racks in the upper and lower thirds of the oven and preheat to 450˚. Line 2 baking sheets with foil. Toss the green beans, 2 tablespoons olive oil, 1 teaspoon salt and the pepper on one baking sheet. Brush some olive oil on the second baking sheet.
Roast the green beans on the lower oven rack until starting to brown and soften, 10 to 15 minutes. Add the garlic and toss. Continue roasting until the garlic is soft and the green beans are lightly charred, 15 to 20 more minutes.
Meanwhile, combine the cracker crumbs, parmesan, paprika and remaining 1 tablespoon olive oil in a shallow bowl, tossing to coat the crumbs with oil. Stir the mayonnaise and mustard together in a small bowl. Lay the fish on the oiled baking sheet. Spread the mayonnaise mixture over the fish, then press the fish into the crumbs to coat. Press any remaining crumbs onto the fish once they are all coated. Return the fish to the baking sheet, crumb-side up.
Roast the fish on the upper oven rack until just cooked through, about 15 minutes. Turn on the broiler and broil until the crumbs are browned and crisp, about 2 more minutes. Serve the fish with the green beans. Sprinkle with parsley and serve with lemon wedges.