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 dill, mint, oregano, 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 Whole-Wheat Pitas in half. Schmear each half generously with the Pickled Feta Spread. Top one half with a burger, then tomatoes, sliced onions and spinach. Place the other pita half on top.
In a small bowl, combine the warm water, sugar and yeast and let sit for 15 minutes.
In the bowl of a stand mixer, combine the flours, cumin, salt and cayenne. Add the yeast mixture and the olive oil. Mix on medium speed to combine and knead the dough. It should take 6 to 7 minutes. The dough will be firm and not sticky or tacky.
Place the dough in a lightly oiled mixing bowl, cover with plastic wrap and let rise in a warm place until the dough has doubled in size, about 1 1/2 hours.
Preheat the oven to 500 degrees F.
Divide the dough into 8 equal pieces and roll into 7-inch rounds (1/4-inch thick). Divide the dough between 2 sheet trays, cover them loosely with a clean tea towel or plastic wrap and let rest for 20 minutes.
Bake the pitas for 3 to 4 minutes, then turn the pitas over and bake for 2 more minutes. Let cool and use as desired.
Pickled Feta Spread:
Place the vinegar, sugar, salt, tea bags, bay leaf and herb stems, if using, in a small saucepan along with 1 cup water. Heat until the sugar and salt has dissolved. Remove from the heat and let cool.
Place the feta in a small container. Add the vinegar mixture and be sure that the feta is submerged in the liquid. Let sit for a couple of hours outside the refrigerator if using right away, or in the refrigerator if using in the future.
Strain the feta from the pickling liquid and discard the tea bags, bay leaf and stems. Place the feta in a food processor and add the yogurt. Pulse until combined.