Prep the buns. Lightly brush both sides of the buns with melted butter and set aside.
Prep the burgers. Divide the beef into four equal sections and use your hands to gently shape each section into a ball. Use the palm of your hand to press each ball into a burger shape. It doesn’t have to be perfectly round. Press your thumb into the center of each burger to prevent doming.
Season. Mix together the salt, pepper, and garlic powder in a small bowl. Sprinkle both sides of the burgers with the seasoning mixture.
Fry the burgers. Heat a cast iron skillet over medium high heat and add just enough vegetable oil to coat. When the oil shimmers, place the patties in the skillet. Fry about 3 minutes, flip, then fry an additional 2-3 minutes or until internal temperature reaches desired doneness (see temp chart below). Place a slice of mozzarella on the burgers for the last minute and cover with a lid or sheet pan to allow it to melt. Remove the burgers to a plate and loosely tent with foil.
Toast the buns. Lower heat to medium low, add the buttered buns to the skillet, and grill until well browned. Remove to a plate.
Make the pesto mayo. While the buns grill, mix the pesto and mayonnaise together in a small bowl.
Assemble. To assemble the burgers, spread both sides of the buns with pesto mayo, then top with burgers, sun dried tomatoes, and arugula. Serve immediately.