For Chile-Lime Butter:
Stir together the butter, shallot, lime zest, lime juice, chile, and salt in a bowl. Set aside. (Roll in parchment and chill if making ahead).
Pat the fish dry and season with salt and pepper to taste on both sides of fish. Dust with flour on both sides. Heat 2 teaspoons of oil in a large nonstick skillet over moderately high heat until just smoking, then sauté 3 pieces of fish, turning over once with a spatula, until golden and just cooked through, 4 to 5 minutes, and transfer to a plate. Sauté the remaining fish in remaining 2 teaspoons of oil in same manner.
Serve each piece of fish with a dollop of Chile-Lime Butter.
Cooks' note: Chile lime butter can be made 1 day ahead and chilled, covered. Bring to room temperature before using.