Preheat the grill to medium-high.
In a large mixing bowl, combine the egg, feta, almond flour or meal, oregano, garlic, salt, and pepper. Add the meat and use your hands to combine, being careful not to overwork the meat. Form into 4 equal-size patties about 1 inch thick.
Grill the burgers, flipping once, until a thermometer inserted in the center registers 160°F and the meat is no longer pink, 3 or 4 minutes per side. In the last minute, warm the pitas or buns on the grill (if using).
Split the pitas or buns open, stuff each with a burger, and top (if desired) with a spoonful of Avocado Cream, Chipotle Hummus, or Don’t Get Beet Hummus.
For a gluten-free alternative to buns, serve the burgers between 2 slices of grilled eggplant rounds. Simply slice 1 large eggplant into 1-inch-thick slices, brush with olive oil, sprinkle with sea salt, and grill for 3 to 4 minutes per side.