
In a medium pot, cover the potatoes with 3 inches of heavily salted, cold water. Bring to a boil and cook until knife-tender, 10–12 minutes. Drain.
Add the potatoes to a large bowl. Mash with a fork or potato masher. Add the tuna, 1 cup of the breadcrumbs, egg, shallot, dill, and black pepper. Mix until combined. Scoop ¼ cup of tuna mixture and shape into a ¾-inch thick disc. Repeat with the remaining tuna mixture.
Toss the tuna cakes in the remaining breadcrumbs.
Line a baking sheet with paper towels. Heat 2 tablespoons avocado oil in a large skillet over medium-high heat. Add 6 tuna cakes and cook until golden brown, 4–5 minutes per side. Transfer to the prepared baking sheet. Repeat with the remaining avocado oil and tuna cakes. Top with dill and serve with lemon wedges and tzatziki sauce, if desired.