Tilapia with Chile Lime Butter

For chile lime butter:
    1/2 stick (1/4 cup) unsalted butter, softened
    1 tablespoon finely chopped shallot
    1 teaspoon finely grated fresh lime zest
    2 teaspoons fresh lime juice
    1 teaspoon minced fresh Thai or serrano chile (preferably red),
      including seeds
    1/2 teaspoon salt

For fish:
    6 (5- to 6-oz) pieces skinless tilapia fillet or farm-raised
      striped bass fillets with skin
    1/2 teaspoon salt
    2 tablespoons vegetable oil

Make chile lime butter:
Stir together butter, shallot, zest, lime juice, chile, and salt in
a bowl.

Prepare fish:
If using striped bass, score skin in 3 or 4 places with a thin sharp
knife to prevent fish from curling (do not cut through flesh). Pat
fish dry and sprinkle with salt. Heat 1 tablespoon oil in a 12-inch
nonstick skillet over moderately high heat until just smoking, then
saute 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. Saute remaining fish in remaining tablespoon oil in same
manner. Serve each piece of fish with a dollop of chile lime butter.