Combine the chicken, buffalo sauce, cream cheese, and ranch seasoning in a bowl, then mash and stir with a fork until the mixture is fully blended. It should look thick and easily spreadable, like a very thick dip, and you can taste it now and add cayenne if you want more heat.
Split the filling between the 2 tortillas and spread it out in an even layer. Cover the surface almost all the way to the edge, but leave a small border so the filling stays inside when you roll.
Scatter the shredded cheddar over each tortilla and roll each one up tightly. Roll as tightly as you can without tearing the tortilla so the slices hold their shape better.
Trim off the uneven ends with a sharp knife, then cut each rolled tortilla into 5 pieces. A clean, sharp blade helps you slice without squishing the filling out the sides.
Place the pinwheels in an even layer in an air fryer basket that has been lightly spritzed with oil spray or lined with parchment, then cook at 370°F for 5 minutes. Look for the edges of the tortillas to turn golden brown and crisp while the centers heat through.
Serve the pinwheels right away. They’re best when the outside is crisp and the filling is warm, creamy, and cheesy.