Combine cilantro, garlic, capers, lime juice, red pepper flakes, 1/4 tsp. each of the salt and black pepper in a blender or mini food processor. Blend, gradually pouring in all but 1 Tbsp. of the olive oil, until mostly smooth. Transfer mixture to a bowl; gently toss with diced avocado.

Heat a grill pan or cast-iron skillet over high heat. Rub tuna steaks with reserved 1 Tbsp. oil; sprinkle with remaining 1/2 tsp. salt and 1/4 tsp. black pepper. Cook the fish for 3 minutes. Turn and cook until desired degree of doneness, 3 minutes longer for medium rare.

To serve, slice tuna steaks and spoon avocado salsa verde overtop.