Fill a large pot with cold water and bring to a boil. Season liberally with salt and add in the gnocchi. Cook it about a minute shy of what the package states and then drain.
Set a large skillet, with high sides, over moderate heat with 2 tablespoons butter. Once melted, add in the drained gnocchi and cook, stirring often, until golden brown and crispy, about 5 to 8 minutes. Use a slotted spoon to transfer the gnocchi onto a plate.
Add the chicken to the hot skillet and cook until golden brown, about 8 to 10 minutes. Season with a pinch of salt and pepper. Transfer chicken to the plate with the gnocchi.
Add another 2 tablespoons of butter to the skillet. Once melted, add the onion, carrot, celery and garlic. Stir and cook until softened, about 5 minutes. Season with a pinch of salt and pepper.
Stir in the hot sauce and heavy cream and bring to a simmer. Return the gnocchi and chicken to the skillet and cook for about 5 minutes longer. Stir in the remaining 2 tablespoons butter to create a smooth and silky sauce.
Remove from the heat and garnish with celery leaves, scallions and blue cheese before serving.