Coat a large saute pan with olive oil, toss in the diced onions and season with salt and crushed red pepper. Bring the pan to medium-high heat and cook the onions for 3 to 4 minutes. Add the garlic and cook for 2 to 4 more minutes. Turn the heat off and let cool.
In a large bowl, combine the lamb, the cooled onion mixture, the oregano, dill, mint, lemon zest and 1/2 to 3/4 cup water. Sprinkle with salt and combine well. Make a little "tester patty." Cook it and taste for seasoning.
Preheat the grill.
Form the lamb mixture into 4 equal patties and sprinkle with salt. Grill the burgers to your desired doneness, 4 to 5 minutes per side for medium-rare. Remove the burgers from the grill and let rest for 3 to 4 minutes.
Cut the pitas in half. Schmear each half generously with the Feta Tzatziki Spread. Top one half with a burger, then tomatoes, sliced onions and spinach. Place the other pita half on top.
Combine the yogurt, feta, vinegar, garlic, mint, dill and cucumbers. Season with salt. Let sit for at least 1 hour at room temperature before serving.