Preheat oven to 400°F. In large pot, melt 1 tbsp butter over medium-high heat and add chicken and cook, stirring occasionally, until chicken is just cooked through, about 5 minutes. Using slotted spoon, transfer chicken to bowl and set aside.
Melt remaining 2 tsp butter in same pot over medium-high heat. Add celery, carrots, and onion and cook, stirring frequently, until vegetables begin to soften, about 5 minutes.
Reduce heat to medium-low. Sprinkle vegetables with flour and cook, stirring, until coated, about 1 minute. Stir in broth and bring to boil over high heat, stirring occasionally. Reduce heat to medium and simmer for 5 minutes. Add peas, cream cheese, thyme, salt, and black pepper and simmer, stirring occasionally, until mixture is thickened and vegetables are just tender, about 8 minutes. Stir in chicken and any pan juices and cook until heated through.
Transfer chicken mixture to 12-by-8- inch baking dish. Unroll dough and press at perforations to seal seams. Top baking dish with dough, letting edges hang over sides of dish (if needed). Poke a few holes in dough to allow steam to vent. Bake until filling is bubbly and crust is golden, about 20 minutes.