Heat oven to 425°F
In 2-quart saucepan, melt butter over medium heat.
Add onion, carrots and celery. Cook until tender.
Place vegetables in a medium bowl.
Add chicken and peas. Toss gently to mix.
Pour the chicken broth into a separate medium pot. Bring to a boil over medium high heat.
Add salt, pepper, thyme and/or rosemary, of your choice, to taste.
Add milk.
Stirring, slowly add enough flour to make a white sauce. Stir all the time, to make sure there are no lumps. The mixture should be thick, but not too thick.
Add chicken and vegetables. Check thickness. Add a bit more flour, to thicken, or a bit of the reserved juice from the chicken, to thin, if desired.
Taste. Adjust seasoning, if needed.
Roll out the uncooked buttermilk biscuit dough to about 1/4 to 1/2-inch thickness.
Choose an oven-safe bowl that will hold all the ingredients.
Turn the bowl over and cut the dough around the shape and size of the bowl (This will be the top of the pie. Be sure to cut as close to the edge of the dough as possible to leave a nice bit for filling the bowl.) Set aside.
Spray the bowl with cooking spray.
Place remaining dough into the bowl and form to fit the bottom and sides. Cut off excess and discard (or roll up and save for making biscuits later. They won't be perfect, but waste not, want not. Store in a tightly sealed container, in the refrigerator, until ready to use.)
Spoon chicken mixture into crust-lined pan.
Top with cut top crust. Seal edge and flute, if desired. Brush with egg white. Cut slits in several places, to vent.
Cook immediately, or store in a tightly sealed container, in the refrigerator, up to a day, until ready to cook.
Place in preheated oven. Bake for 30 - 40 minutes or until crust is golden brown. (Cover crust edges with strips of foil during the last 15 - 20 minutes, or when you see them start to darken, to prevent burning, if desired.)
Remove from oven and let stand 5 minutes before serving.
Serve hot.