Start the broccoli: Put a large nonstick skillet on the stove over medium-high heat. Add 1 tablespoon olive oil and let it heat until it looks loose and shiny, about 30 seconds.

Add the broccoli to the skillet. Spread it into as even a layer as you can and cook without stirring for 2 minutes so it starts to brown on the bottom.

Stir the broccoli. Add 2 tablespoons water to the skillet, cover with a lid or an upside-down sheet pan, and cook for 2 minutes, until the broccoli is bright green and just tender when pierced with a fork.

Uncover the skillet and cook 1 more minute to let the water cook away. Transfer the broccoli to a plate and set it aside.

Mix the cakes: In a small bowl, stir 1/4 cup of the mayonnaise with 1 tablespoon of the lemon juice until smooth. Set this lemon mayo aside for serving.

Put the drained tuna in a large bowl. Break it into small flakes with a fork.

Add the beaten egg, panko, Dijon mustard, the remaining 1 tablespoon mayonnaise, half of the scallions, the parsley, 1 teaspoon lemon zest, and 1 tablespoon lemon juice to the bowl with the tuna.

Mix with the fork until the tuna mixture is evenly combined. It should hold together when you press a spoonful in your hand; if it feels loose, let it sit for 2 minutes so the crumbs can absorb moisture.

Divide the mixture into 8 equal portions. Shape each portion into a cake about 3 inches wide and 1/2 inch thick, then set the cakes on a plate.

Cook the tuna cakes: Wipe out the skillet if any dark bits are left behind. Put the skillet back over medium heat and add the remaining 2 tablespoons olive oil.

When the oil is hot, put 4 tuna cakes into the skillet. Cook them without moving them for 3 to 4 minutes, until the bottoms are deep golden brown.

Turn the cakes with a thin spatula. Cook the second side for 3 to 4 minutes, until browned and the centers feel firm. Transfer the cooked cakes to a clean plate.