Place a medium sauté pan over medium heat. Add 2 teaspoons olive oil and warm through before adding the red onion. Toss the onion in the oil and cook over medium heat until softened, 5 to 7 minutes. Add the garlic, oregano and red pepper flakes and sauté until fragrant, about 30 seconds, before removing from the heat. Allow to cool for 2 minutes.
Place the ground lamb in a bowl and add the onion mix along with the feta and ¾ teaspoon kosher salt and ¼ teaspoon black pepper. Using your hands, gently mix everything together, ensuring the components are well distributed while being careful not to overmix. Divide into 4 balls and shape into burger patties, using your thumb to add a small dimple to the middle of each burger. Set aside until ready to cook.
Toss the eggplant with 1 tablespoon olive oil, ½ teaspoon kosher salt and ¼ teaspoon black pepper.
Both the eggplant and the burgers will cook at the same time, for 5 minutes per side. If cooking on the stove, warm an oiled grill pan over medium heat. Alternatively, heat a grill to medium and cook them on a well-oiled grate over indirect flame. Transfer both the burgers and the eggplant to a plate lined with a paper towel and set aside.
While the burgers and eggplant are cooking, add the mint and parsley to the mayonnaise along with the lemon juice. Whisk until smooth and smear on both halves of each English muffin.
Place a burger on the bottom half of each muffin. Garnish with the grilled eggplant and arugula and add the top half of the muffin. Serve.