Prepare all of the vegetables and slice the hamburger buns. Mix together the mayonnaise and Dijon mustard to create your aioli.
Season the burger patties with salt and pepper on both sides.
Place a heavy-bottomed or cast iron pan over medium heat. Melt the butter in the pan. Place the buns in the pan, interior-side down, and toast until golden brown. Put aside and reserve.
Turn the heat up to medium-high heat. In the same pan, add the burger patties. Cook for about 6-8 minutes on each side, or until cooked medium well. Internal temperature will read 155 degrees. During the last 2 minutes of cooking, add the cheddar cheese and let melt.
Remove patties from pan and place atop burger buns. Assemble ingredients on top of each patty, spread aioli on the bun lid and then place it over the burger. Serve with salad, potato salad or your choice of side.