Place chicken in deep sauce pan with water to cover and season well with salt and pepper. Add celery tops. Bring to a boil.
Once water is boiling, cover pan and turn off heat. Set pan aside for just a minute or two so it doesn’t boil over and then return it to the still warm eye.
The chicken will continue to slowly cook, {poach} as the burner cools, while you prepare vegetables.
Melt 1 T butter in skillet and drizzle with olive oil. Add chopped vegetables and cook over medium, stirring occasionally until softened, about 10 minutes. Add white sauce and cubed cheese.
Remove chicken from broth and shred or chop. If it’s not cooked all the way through, don’t worry, it will finish while enchiladas are baking.
Add chicken to vegetables and sauce and stir. Season with salt & pepper.
Lay flour tortillas out and divide filling evenly, placing several heaping spoonfuls in a line down the center.
Roll up and place seam side down in greased baking dish pushing them closely together.
Cover with grated cheese.
At this point dish can be refrigerated until baking up to 24 hours.
Bake uncovered for 30-45 minutes.
Baking time will depend on temperature of dish before cooking. Cover loosely with foil if necessary if cheese is getting too brown.
Separate enchiladas to serve or cut into squares