Add the asparagus, mushrooms, squash and onion to a sheet pan and toss with 2 tablespoons of the olive oil, 1 1/2 tablespoons of the Italian seasoning, 1/2 teaspoon salt and 1/4 teaspoon pepper. Move to one side of the pan.
Toss the shrimp with the remaining olive oil and Italian seasoning blend and a pinch of salt and pepper. Add to the other side of the pan.
Put the foil-wrapped pita in the oven with the sheet pan. Bake, tossing the shrimp and vegetables halfway through, until the shrimp are just cooked and the vegetables are lightly roasted and starting to turn golden, 12 to 15 minutes. Squeeze over the juice of one of the lemons.
Mix together the yogurt, dill, lemon zest and the juice of the second lemon in a small bowl. Season with salt and pepper.
Lay out the sheet pan with the shrimp and vegetables alongside a plate with the sliced tomatoes and lettuce, warm pita halves and yogurt mixture. Put the olives and feta into separate bowls and have everyone fill their own pita with the different elements. Garnish with additional dill.