Make the mayonnaise topping by combining the mayonnaise, fresh lemon juice, salt, pepper, and finely chopped basil.
To the ground turkey, add the chopped sundried tomatoes, minced garlic, salt, and black pepper. Mix with your hands until combined.
Divide the ground turkey mix into 4 equal portions and shape them with your hands into patties (slightly bigger than your burger buns as the patties will shrink when cooking).
To a cast iron pan (or grill pan or frying pan/skillet) over medium heat, add the oil. When the oil is hot, add the patties, and cook for about 5 minutes per side until fully cooked. The turkey patty is fully cooked when the internal temperature, when measured with a meat thermometer, is 165°F (or 74°C). After cooking it on the first side for about 5 minutes, flip and add the cheese slices on top to melt. To make sure the cheese melts better, you can cover the pan with a lid, but this step is optional.
Toast the buns in a pan for a couple of minutes until golden. If using brioche buns, you don’t need additional butter or oil as the buns have a higher content of butter/fat. But you can optionally use some if you want a more indulgent toasted bun!
Prepare your turkey burger assembly board: cooked turkey patties with melted cheese, toasted buns, avocado (cut in half and sliced), sliced tomatoes, arugula, and the lemon-basil mayo.
Assemble the turkey burger, spreading some of the lemon-basil mayo on the inside of each half of the toasted bun and then adding the cooked turkey patty with melted cheese, a couple of slices of tomato, a fourth of an avocado (sliced thinly), and a handful of arugula.