Preheat the air fryer to 400°F for 3 minutes. While it heats, take the frozen breaded chicken patties out of the packaging - no need to thaw them.
Place the frozen chicken patties in a single layer in the air fryer basket, making sure they are not overlapping. Give the tops a light mist of olive oil spray - this helps them develop an extra-crispy golden crust.
Air fry at 400°F for 10 to 12 minutes, flipping the patties over at the halfway mark. They should be deep golden brown and crispy on both sides, with an internal temperature of at least 165°F.
During the last minute of cooking, lay a slice of pepper jack cheese on top of each patty. Close the air fryer and let the residual heat melt the cheese until it's perfectly gooey.
While the patties cook, stir together the mayonnaise, sriracha, and honey in a small bowl until smooth. Taste and adjust the heat to your liking.
Split the brioche buns and place them cut-side up in the air fryer. Toast at 400°F for 1 to 2 minutes until lightly golden and warm.
Spread a generous layer of sriracha mayo on both halves of each toasted bun. Place a cheesy chicken patty on the bottom bun, then top with shredded lettuce, a tomato slice, and dill pickle chips. Close the sandwich and serve immediately while everything is hot and crispy.