Add the 5 smashed garlic cloves, diced large yellow onion, 3 tablespoons lemon juice, 1/2 cup low-sodium chicken broth, 18–24 ounces boneless skinless chicken breasts, 1 tablespoon dried oregano, 2 teaspoons dried mint, and 1 teaspoon kosher salt into the Instant Pot.
Secure the lid, set the valve to sealing, press the "Poultry" button, press "Adjust" to select high pressure, and set the time to 25 minutes.
Allow the Instant Pot to come to pressure and cook; when the 25-minute cook time finishes carefully quick-release the steam according to your Instant Pot's instructions.
Open the lid, remove the cooked chicken to a cutting board if desired, then shred with two forks; return shredded chicken to the pot and stir to combine with the cooked onions and juices, keeping it warm.
Make the cucumber: place the 1/2 cup shredded cucumber in a small bowl, sprinkle with the kosher salt, and let rest at least 5 minutes. Then squeeze out excess liquid.
In another 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 until smooth, then fold in the drained shredded cucumber to make the gyro sauce.
Assemble gyros by placing about 1/4 cup shredded chicken and 1–2 tablespoons of the gyro sauce on each pita, then top with halved cherry tomatoes, sliced red onion, shredded romaine, and feta cheese.
Serve immediately.