In a mixing bowl, combine the lamb, shallots, mint, garlic, salt, cumin, allspice, 1/4 teaspoon of the cayenne, and the cinnamon and mix gently but thoroughly to combine. Using your hands, shape the mixture into 6 wide patties about 1/2-inch thick and transfer to a large plate or a platter. Cover with plastic wrap and refrigerate for at least 2 hours and up to overnight.

Make the feta dip: Combine the remaining 1/8 teaspoon cayenne pepper, feta cheese, cream cheese, mayonnaise, green onions, 1 teaspoon of the olive oil, and lemon zest in a mixing bowl and stir to blend well. Cover with plastic wrap and refrigerate for at least 1 hour and up to overnight to allow flavors to blend.

When ready to cook the burgers, preheat a grill to medium-high. Brush the burgers on both sides with the remaining tablespoon of olive oil. When the grill is hot, add the burgers and cook for about 4 minutes on each side for medium. Transfer the burgers to a platter and cover loosely with foil. Place the buns, cut sides down, and in batches if necessary, in the hot skillets and cook until warmed through and slightly toasted, about 2 minutes. Serve the buns with the cooked burgers and spoon some of the feta spread over each burger. Garnish with lettuce, tomatoes, and sliced roasted peppers if desired, and serve immediately.

In a mixing bowl, combine the lamb, shallots, mint, garlic, salt, cumin, allspice, 1/4 teaspoon of the cayenne, and the cinnamon and mix gently but thoroughly to combine. Using your hands, shape the mixture into 6 wide patties about 1/2-inch thick and transfer to a large plate or a platter. Cover with plastic wrap and refrigerate for at least 2 hours and up to overnight.

Make the feta dip: Combine the remaining 1/8 teaspoon cayenne pepper, feta cheese, cream cheese, mayonnaise, green onions, 1 teaspoon of the olive oil, and lemon zest in a mixing bowl and stir to blend well. Cover with plastic wrap and refrigerate for at least 1 hour and up to overnight to allow flavors to blend.

When ready to cook the burgers, preheat a grill to medium-high. Brush the burgers on both sides with the remaining tablespoon of olive oil. When the grill is hot, add the burgers and cook for about 4 minutes on each side for medium. Transfer the burgers to a platter and cover loosely with foil. Place the buns, cut sides down, and in batches if necessary, in the hot skillets and cook until warmed through and slightly toasted, about 2 minutes. Serve the buns with the cooked burgers and spoon some of the feta spread over each burger. Garnish with lettuce, tomatoes, and sliced roasted peppers, if desired, and serve immediately.