Add the smashed garlic, diced yellow onion, 3 tablespoons lemon juice, 1/2 cup low-sodium chicken broth, chicken breasts, 1 tablespoon dried oregano, and 2 teaspoons dried mint to the Instant Pot.
Secure the lid and set the valve to sealing. Select the Poultry setting, press Adjust to choose high pressure, and set the cook time to 25 minutes.
Allow the Instant Pot to come to pressure and cook; once the 25 minutes finish, carefully quick-release the steam using the valve until the pressure is fully released, then open the lid.
Remove the chicken or leave it in the pot and use two forks to shred the breasts; stir the shredded chicken into the cooked onions and pan juices to keep it moist and warm.
For the sauce, place the shredded cucumber in a small bowl and sprinkle with 1 teaspoon kosher salt; let sit at least 5 minutes, then squeeze out excess liquid.
In a mixing bowl combine 1/2 cup plain Greek yogurt, 2 minced garlic cloves, 2 teaspoons lemon juice, 2 teaspoons olive oil, and 3 teaspoons dry dill; stir to combine, then fold in the drained cucumber.
Assemble each pita with about 1/4 cup shredded chicken and 1–2 tablespoons of the gyro sauce, then top with cherry tomatoes, red onion strips, shredded romaine, and crumbled feta.
Serve immediately.