If your chicken isn't shredded, use two forks to pull it into bite-sized pieces; measure so you have about 1.25 lb shredded total. Pat the chicken dry so it absorbs sauce more evenly.
Make the buffalo sauce: melt 2 tablespoons butter in a small skillet over medium-low heat, whisk in 1/2 cup hot wing sauce and 1 teaspoon smoked paprika until smooth, then remove from heat. Taste and season with 1/2 teaspoon kosher salt as needed.
Heat a large nonstick skillet over medium. Add the shredded chicken and pour the warm buffalo sauce over it. Toss continuously in the skillet for 3 to 4 minutes so the chicken warms through and the sauce clings to the meat.
Stir in half (1/2 cup) of the shredded cheddar into the sauced chicken, cooking just long enough to let the cheese soften and help bind the filling, about 1 minute. Remove from heat.
Warm the tortillas so they’re flexible: wrap them in a damp paper towel and microwave 25–30 seconds, or heat each briefly in a dry skillet for 10–15 seconds per side.
Assemble the burritos: lay a warm tortilla flat. In a line down the center place about 1/4 of the buffalo chicken, a handful of shredded lettuce, 2 tablespoons chopped celery, a sprinkle of green onion and a drizzle (about 1 tablespoon) of ranch. Top with a little of the remaining cheddar.
Fold the sides of the tortilla in, then roll tightly from the bottom to form a burrito. Press the seam to close.
Heat 1 tablespoon butter or oil in a skillet over medium. Place each burrito seam-side down and cook 2–3 minutes until golden, then flip and toast the other side 2–3 minutes until crisp and warm throughout.
Let the burritos rest 2 minutes after toasting (this helps them hold together), slice in half if desired and serve with extra ranch for dipping.