Whisk flour, salt, and sugar in a large bowl. Add cold cubed butter and work into the flour until rough crumbs form with pea-sized butter pieces visible. Add ice water one tablespoon at a time until dough holds together when squeezed. Divide into two disks, wrap in plastic, and refrigerate at least 30 minutes.
Place chicken thighs in a Dutch oven and cover with broth. Bring to a gentle simmer over medium heat, partially covered, for 20 to 25 minutes until the thickest piece reaches 74°C (165°F). Remove chicken, reserve the broth. Discard skin and bones; shred meat into pieces.
In the same Dutch oven, melt butter over medium heat. Add onion and cook 5 to 6 minutes until soft. Add carrots and celery for 3 minutes. Add garlic and stir 60 seconds.
Sprinkle flour over the vegetables and stir to coat. Cook for 90 seconds until the raw flour smell fades. Pour reserved broth back in gradually while stirring. Simmer 4 to 5 minutes until the filling thickens to a heavy coating consistency.
Stir in heavy cream, thyme, rosemary, shredded chicken, and frozen peas. Season with salt and pepper. Transfer filling to a bowl and cool completely to room temperature, about 20 minutes.
Preheat oven to 200°C (400°F). Roll one pastry disk to a 30cm circle and press into the pie dish without stretching. Add cooled filling. Roll the second disk and lay over the filling. Seal and crimp the edges.
Brush the top crust and crimped edges with egg wash. Cut 4 to 5 venting slits across the top. Chill the assembled pie 15 minutes in the refrigerator.
Bake on the center rack for 30 to 35 minutes until the crust is deep amber and the filling bubbles at the vents. Rest on a wire rack for 10 minutes before slicing.