Toss together 12 oz. Monterey Jack, coarsely grated (about 3 cups), ¼ cup crumbled blue cheese, and 1 Tbsp. plus 1½ tsp. cornstarch in a large bowl with your hands until cheese is evenly coated. Set aside.

Generously season 1½ lb. skinless, boneless chicken thighs (about 6) or breasts (4–6) with kosher salt and freshly ground pepper. Heat 2 Tbsp. extra-virgin olive oil in a medium Dutch oven or other heavy pot over medium-high. Working in 2 batches if needed, arrange chicken in a single layer and cook until deeply browned underneath, 6–8 minutes. Turn and cook until deeply browned on other side and cooked through, 6–8 minutes. Transfer chicken to a plate. Reserve pot (no need to wash it; some brown bits stuck to the bottom are okay). Let chicken cool slightly, then shred.

Meanwhile, combine 12 oz. penne or other medium pasta, ½ cup (1 stick) unsalted butter, 2 tsp. Diamond Crystal or 1¼ tsp. Morton kosher salt, and 4½ cups water in reserved pot and bring to a boil. Cook, stirring often to keep pasta from sticking, until about two thirds of water has been absorbed and pasta is very al dente, 6–8 minutes for most pasta shapes (about 2 minutes less than package directions).

Add ½ cup Buffalo wing sauce to pasta and cook, stirring often, until about half of hot sauce has been absorbed and pasta is al dente, about 2 minutes. Reduce heat to low and add reserved cheese mixture a handful at a time, stirring after each addition until mostly melted before adding more. Stir until mixture is creamy, then mix in shredded chicken. Remove pot from heat. Taste pasta; season with more salt and pepper and add more Buffalo sauce if needed.

Divide pasta among shallow bowls and top with crumbled blue cheese, thinly sliced scallions, and thinly sliced celery stalks and leaves if desired.