Slice the tuna steak into 2 or 3 long rectangular portions. Season each side of each slice generously with salt and pepper.
Place the cilantro and parsley in a small food processor (see Note). Pulse to chop finely. Add the lime zest, lime juice, tamari, sesame oil, garlic, and ginger. Pulse several times to combine. Scrape down the sides of the bowl.
With the food processor running, slowly pour in ¼ cup of olive oil. Scrape down the sides again and pulse a few times. If the sauce is too thick, add more oil until the sauce reaches desired consistency.
In a large skillet, heat the avocado oil over medium-high heat until quite hot. Gently place the tuna in the oil and sear for 1 minute without moving. Sear each side in same manner. The tuna will still be pink in the middle, or cook slightly longer for more well-cooked fish.
Remove the fish to cutting board and slice approximately ½ inch (13 mm) thick.
Drizzle the tuna generously with the dressing and serve.