Place burrata on a plate and freeze for 15 minutes.
Meanwhile, spoon flour into a medium shallow bowl. Whisk egg in a second medium shallow bowl. Stir together breadcrumbs, thyme, and salt in a third medium shallow bowl.
Carefully dredge cold burrata in flour, making sure you don’t break it open; dip in egg, letting excess drip off. Dredge in breadcrumb mixture, patting to adhere. Transfer coated burrata back to a plate and freeze for 15 minutes.
Generously coat air fryer basket or tray with cooking spray. Place dredged burrata in air fryer basket; generous coat with cooking spray. Air-fry burrata at 350°F until burrata is golden brown, 5 to 7 minutes.
Spoon pesto in a serving bowl. Carefully transfer cooked burrata on top of the pesto and break open. Serve burrata alongside toasted ciabatta.