Tip the tuna into a fine-mesh sieve and press with the back of a spoon for about 1 minute, until it stops dripping. Flake it into a large bowl and add the eggs, 80 g shredded cheddar, panko, mayonnaise, celery, onion, mustard, lemon juice, parsley, garlic powder and pepper. Mix with a fork until a handful holds together when squeezed.
Divide the mix into 8 portions of about 95 g (1/3 cup) each. Shape each into a patty about 9 cm / 3.5 inches wide and 2 cm / 3/4 inch thick and set them on a parchment-lined sheet pan 33 x 46 cm / 13 x 18 inch. Chill for 15 minutes and heat the oven to 95 C / 200 F while you wait.
Heat 1 tbsp oil in a 30 cm / 12-inch skillet over medium heat for 2 minutes, until it shimmers. Add 4 patties and cook for 3 minutes, until the underside is deep golden. Flip once with a wide spatula and cook 3 minutes more.
Top each patty with a heaping tablespoon (10 g) of the remaining cheddar. Cover the skillet for 1 minute, until the cheese melts. Check one patty with a thermometer for 165 F / 74 C in the center, then move the batch to a second sheet pan in the warm oven.
Add the remaining 1 tbsp oil and cook the other 4 patties the same way. While this batch cooks, toast the muffin halves for 2 to 3 minutes, until golden, and slice the tomatoes.
Assemble each serving open-faced: a toasted muffin half, a tomato slice, then a patty. Serve 2 per person right away while the cheese is soft.