Get out and measure your ingredients.
Melt butter in large saucepan over medium heat. Add onion, carrots, celery, salt, and pepper and cook until vegetables begin to soften, about 6 minutes.
Add flour and cook, stirring constantly, until golden, 1 to 2 minutes. Slowly stir in broth and half-and-half and bring to boil over medium-high heat.
Stir in potato and thyme. Reduce heat to medium and simmer
Simmer until sauce is thickened and potato is tender, about 8 minutes.
Off heat, stir in chicken and corn or peas, if using.
Transfer filling to dough-lined pie plate.
Add filling and press until as evenly distributed as possible.
Loosely roll remaining dough round around rolling pin and gently unroll it onto filling. Trim overhang to ½ inch beyond lip of plate. Pinch edges of top and bottom crusts firmly together. Tuck overhang under itself; folded edge should be flush with edge of plate. Crimp dough evenly around edge of plate using your fingers.
Cut four 2-inch slits in top of dough.
Brush top of pie with egg.
Place in smoker and smoke for 90 mins – 2 hours at 250 degrees.
Checking pie about every 30 minutes to check doneness.
Remove from smoker and let cool on a wire rack for 45 minutes before serving.
Slice.
Serve.
Enjoy every bite!