Combine the grated Monterey Jack, crumbled blue cheese, and cornstarch in a large bowl. Toss the cheese gently to coat it in the cornstarch. This step ensures the cheese melts evenly into the sauce later and creates a velvety finish.
Season your chicken thighs or breasts with salt and black pepper on both sides. Heat olive oil in your Dutch oven over medium-high heat. Add the chicken to the pot, giving each piece enough room to sear without overcrowding. Cook each side for about 6-8 minutes until golden brown, then transfer to a plate. Once cooled slightly, shred the chicken into bite-sized pieces.
Using the same pot, add your pasta, butter, salt, and 4½ cups of water. Stir occasionally to prevent the pasta from sticking. Cook on medium heat until most of the water is absorbed and the pasta is just shy of al dente (about 6-8 minutes).
Pour in the Buffalo wing sauce, stirring consistently to coat the pasta evenly. Allow it to simmer for a couple of minutes to intensify the flavor. Reduce the heat to low and start adding the cheese mixture, one handful at a time. Stir after each addition until melted and silky.
Once the sauce is perfectly creamy, toss in the shredded chicken. Stir well to combine everything and allow the chicken to warm through. Taste the dish, and adjust seasoning or add more Buffalo sauce to suit your spice preference.
Divide the pasta into bowls, sprinkling a little extra blue cheese, scallions, and celery leaves if desired. Serve it hot and enjoy every cheesy, spicy bite!