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Brush the chicken breasts with olive oil and season with the dried thyme, salt and freshly ground black pepper.
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Air-fry chicken breasts at 380°F for 10 to 12 minutes, flipping over halfway through the cooking process.
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Transfer cooked chicken to a cutting board and cut into ½-inch cubes. Place the chicken cubes in a bowl, add the sun-dried tomatoes and fresh oregano. Toss to combine and set aside to cool.
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Lightly coat your countertop or a silicone mat with a little flour. Roll the pizza dough out into a 10x18-inch rectangle. Cut the dough in half lengthwise and then cut each half in 3 pieces. You should have 6 rectangles of dough. Brush the beaten egg around the edges of each rectangle.
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Spread about one tablespoon of the pesto sauce in the middle of each dough rectangle leaving the egg-washed boarder uncovered. Place a slice or two of the mozzarella cheese on top of the pesto and then top with the chicken mixture, dividing it equally between the 6 rectangles.
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To make the pocket, first fold the short edges of the rectangle over the filling and then fold the long edges over the filling, overlapping and pinching to seal the dough together. Lightly oil the air fryer basket and place the pocket, seam side down into the basket. Brush the top of the pocket with the remaining egg wash or a little olive oil. Sprinkle some Italian seasoning on top. Using a fork, gently pierce 3 sets of holes across the top of the pocket to vent the dough. (At this point you can freeze the pockets and then store in a zipper sealable plastic bag for a future snack.)
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Air-fry at 350°F for 10 minutes and then flip the pockets over and air-fry for an additional 5 minutes. OR bake at 400°F for 20 to 25 minutes, until the outside is lightly browned.
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Transfer to a serving plate or cutting board. Remember they are hot pockets, so let them rest for a few minutes before digging in!