Cook the Tuna: Preheat a medium skillet over high heat. Add 1 teaspoon of avocado oil. Season the tuna with sea salt and freshly cracked black pepper, then add it to the pan. Brown the tuna on both sides, then reduce the heat and continue cooking until the tuna is cooked through (internal temperature of 145˚).
1-inch thick steak: 3–4 minutes per side1 ½-inch thick steak: 5–6 minutes per side2-inch thick steak: 8–10 minutes per side, plus 2–3 extra minutes over indirect heat if needed Make the Pesto: Place the 3 tablespoons lemon juice, ¼ cup extra virgin olive oil, 1.25 ounces of basil, 1 teaspoon grated garlic, 1 tablespoon roasted pine nuts, ½ cup pecorino cheese, salt and pepper in a food processor.
Pulse until the pesto is well combined. Store in an airtight container in the refrigerator for up to 3 days.
Make the Tuna Salad: Once the tuna has cooled, flake it apart with your hands. Place it in a bowl and mix it with a 2 tablespoons red onion, 1/4 cup red pepper, 1/2 cup mayonnaise and the pesto. Season with salt and pepper and mix to combine. Store in a refrigerated airtight container for up to 3 days.
Make the Sandwich: Brush the 2 buns with 1 tablespoon room temperature butter and place them cut side down on a griddle to lightly toast them.
Preheat the broiler.
Next place the buns on a baking sheet, toasted side up. Divide the provolone between the two top buns. On the bottom buns, divide the tuna salad. Place under the broiler, place the baking tray 3 to 6 inches below the broiler, and cook until the cheese starts to brown, and the tuna is warmed through, about 6 minutes.
Remove from the broiler and serve immediately.