Adjust rack to top position and preheat oven to 425°F.
Place tomato rounds on a parchment-lined sheet pan. Sprinkle with ½ tablespoon fresh oregano (double this amount if doubling the recipe), Kosher salt and freshly ground black pepper. Roast for 20 minutes, until tender.
Meanwhile, pierce the bag of spinach a few times with a sharp knife. Microwave for 1 minute and then let cool in bag for 5 minutes. Place spinach in a clean towel and squeeze out any excess liquid. Roughly chop and transfer to a large mixing bowl.
To the bowl with the cooled spinach, add the red onion, garlic powder, cumin, and a generous amount of Kosher salt (I used 1 ½ teaspoons for 5 burgers), freshly ground black pepper, and remaining oregano. Mix well to combine, then add in the ground turkey and feta and gently toss to coat.
Divide mixture into 5 equal portions and form into patties (It helps if you wet your hands first since the ground turkey is sticky). Press down to create a dimple in the center. At this point you can cover and refrigerate the patties overnight. Let come to room temperature before cooking.
Stovetop or Flattop Cooking Method (this is the method I prefer): Heat a large drizzle of avocado oil in a large pan over medium-high heat. Add the turkey burgers and cook for 4 minutes per side. Cover the pan with a lid for the last 2 minutes (this helps the burgers cook through). If using a flattop, cover the burgers with a large pan lid as well.
Outdoor Grill Cooking Method: Heat a clean grill over medium-high heat. Grease your grill or rub with the cut side of a potato to make it nonstick (I like to do both since ground turkey is extremely sticky). Spray both sides of the burgers with cooking spray. Cook burgers for 5-6 minutes per side, until browned and cooked through.
Fill the hamburger buns with the aioli, burgers, lettuce, and roasted tomatoes and serve immediately.