Preheat the oven to 300 degrees F.
For the gyro meatloaf: Gather the minced onions in a clean kitchen towel and wring out as much liquid as possible. In a large mixing bowl, combine the onion, beef, lamb, milk, breadcrumbs, oregano, thyme, fennel, coriander, egg and some salt and pepper and mix together. Fry up a small test batch; taste for salt and pepper, and adjust the seasoning as needed.
Transfer the meatloaf mixture to a food processor and process it to the consistency of a paste. Tightly pack it into a loaf pan, making sure there are no cracks or air pockets. Bake until cooked through, about 45 minutes. Let cool completely before cutting into thin slices.
For the tzatziki sauce: Grate or mince the cucumber and add it to a bowl along with the yogurt, dill, lemon juice, mint, olive oil, coriander and cumin. Whisk to blend. Season with salt and pepper.
For the lemon vinaigrette: In a blender, combine the lemon juice, oregano, thyme and mustard. With the motor running, add the olive oil in a slow, steady stream to emulsify.
For the salad: Toss the lettuce, red onions and tomatoes with about 4 tablespoons of the vinaigrette.
Heat the 1 teaspoon olive oil in a large saute pan over high heat. Briefly warm up the pitas in the pan, giving them just long enough on each side to get hot but not crispy. In the same pan, sear the slices of gyro on both sides, about 2 minutes per side.
Spread 1 heaping tablespoon of tzatziki sauce onto each pita. Top with some slices of meatloaf and finish with some of the salad. Roll the pitas up tightly and wrap them in foil or secure them with a toothpick. Serve.