Adjust oven rack to middle position and heat oven to 450 degrees. Line rimmed baking sheet with parchment paper. Unroll dough on baking sheet. Use 12-ounce ovensafe ramekin as guide to cut out 2 dough rounds about 1/2 inch larger than mouth of ramekin. Fold under and crimp outer 1/2 inch of dough, then cut 3 vents in center of each crust. Bake until crusts just begin to brown and no longer look raw, about 7 minutes; set aside.
Meanwhile, melt butter in medium saucepan over medium heat. Add carrots, onion, celery and 1/2 teaspoon salt and cook until vegetables are softened and browned, 8 to 10 minutes. Stir in garlic and thyme and cook until fragrant, about 30 seconds. Stir in flour and cook for 1 minute.
Slowly whisk in broth, cream, and soy sauce, scraping up any browned bits. Nestle chicken into sauce and bring to simmer. Cover, reduce heat to medium-low, and cook until chicken registers 160 degrees, 10 to 15 minutes. Transfer chicken to plate; let cool slightly. Using 2 forks, shred chicken into bite-size pieces.
Meanwhile, return pan with sauce to medium heat and simmer until thickened and sauce measures 2 cups, about 5 minutes. Off heat, return shredded chicken and accumulated juices to pan. Stir in peas, parsley, and lemon juice and season with salt and pepper to taste.
Divide filling between ramekins and place parbaked crusts on top of filling. Place pot pies on baking sheet and bake until crusts are deep golden brown and filling is bubbling, 10 to 15 minutes. Let pot pies cool for 10 minutes before serving.