To make the roasted red pepper aioli, place the mayonnaise, roasted red pepper and garlic in a food processor and blend until smooth. Add the olive oil while the processor is still going, but stop once the texture is smooth. Salt and pepper to taste. Store the aioli in the refrigerator until ready to serve (use it on your turkey sandwich tomorrow too!).
Preheat the grill to high heat.
Form your hamburger patties with a dimple in the middle to keep them from blowing up like the Michelin Man. Here are some tips on making the burger patties.
Once the grill is nice and hot (preheat for around 15 minutes), place the hamburger patties on the hottest part of the grill and leave the lid open.
Grill for around 6 minutes (these times vary per grill, but 6 - 8 minutes is about right) on the first side.
Flip the burgers and put the feta cheese on the patties to heat up while the burgers cook the rest of the way. Note, the feta will not melt so technically you can add it at the end or on the platter. I like to put it on the burgers to at least get it warm while the burgers continue cooking. It is up to you.
Grill for another 6 minutes on this new side of the patty or until a good meat thermometer reads 155 to 160 degrees.
Remove the burgers from the grill and toast some buns on the still hot grill.
Serve the feta burgers on a toasted bun with a generous amount of the roasted red pepper aioli on top!