For the onions: In a saute pan over medium heat, add the butter and olive oil. When the butter starts to foam, add the onions, salt and sugar. Cook until the onions break down and are caramel in color, 25 to 30 minutes.
While the onions cook, start the "Mayoli": In a bowl, whisk together the mayonnaise, olive oil, dried basil and granulated garlic. Refrigerate 30 minutes.
Combine the lamb loin, granulated garlic, salt and pepper in a food processor and chop until completely ground. Divide the meat into 6 balls and then form into patties. In a saute pan over medium-high heat, add 2 tablespoons of the olive oil and cook 3 of the burgers, about 3 minutes per side.
Remove and keep warm on a plate under foil.
Add the remaining 2 tablespoons olive oil and cook the remaining 3 burgers, about 3 minutes per side.
Butter the pita or rolls and toast in a skillet until golden brown, about 2 minutes. To serve, spread 1 pita with "Mayoli", top with a lamb burger, some onions and 1/2 ounce of the feta. Close the burger with another pita. Continue until all 6 burgers are built.
