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).

For fish:

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.