Combine ground turkey, salt, and pepper in a large bowl. Mix with a spatula until the spices are blended. Add onion and panko breadcrumbs. Gently stir a few times until the ingredients are well combined. Do not stir too much or the burger patties will become tough after cooking.
Divide the ground turkey mixture into 4 equal parts (or 3 if making thick burgers). Shape each portion into a ball by hand. Press to shape a round, 1/2-inch-thick disc (1 inch thick if making 3 burgers). Place burger patties on a plate and set aside.
Heat oil in a large skillet over medium-high heat until hot. If using an electric stove, turn on another element to low heat.
Add burger patties carefully into the skillet and lightly press them. Add an oil splatter screen (if using). Cook for 3 to 4 minutes, until the bottom side turns dark brown. Flip and cook the other side for 1 minute. Transfer the skillet to the low heat element (or turn to low heat). Cover and cook for 3 to 5 minutes (more time for thicker patties), until the turkey is cooked through or a thermometer inserted to the thickest part reads 165 degrees Fahrenheit.
Flip the burgers. Place cheese on top. Cover and cook until the cheese melts, about 1 minute. Transfer to a plate to cool.
Toast the burger bun with butter or olive oil in a separate skillet until charred.
To assemble burgers, spread avocado (or mayonnaise) on the burger bun, add pickles, and tomato, top with turkey patty and the other half of the burger bun. Serve warm.