Put the parsley, capers, garlic, lemon juice, anchovy paste, mustard, 1/2 teaspoon of the salt, and 1/8 teaspoon of the pepper into a food processor. Pulse to chop, six to eight times. With the machine running, add the 1/2 cup oil in a thin stream to make a coarse puree. Leave the sauce in the food processor and, if necessary, pulse to re-emulsify just before serving.
Heat a grill pan or heavy cast-iron pan over moderately high heat. Rub the tuna steaks all over with the 1 tablespoon oil.
Sprinkle with the remaining 1/4 teaspoon salt and 1/8 teaspoon pepper. Cook the fish for 3 minutes. Turn and cook until done to your taste, 3 to 4 minutes longer for medium rare. To serve, top the tuna steaks with the avocado and drizzle with the sauce.
Fish Alternatives: Firm fish steaks, such as sturgeon, mako shark, and swordfish, all make excellent alternatives to the tuna. You'll want to cook them longer until just done, rather than medium rare.
Wine Recommendation: The brazen flavors of anchovy paste and capers make a white wine with lots of acidity a must. A Sancerre or a lighter-styled Muscadet de Svre-et-Maine, both from the Loire Valley in France, are good examples.
