Preheat oven to 425 degrees F (220 degrees C). In a large bowl, whisk together the flour, baking powder, sugar, and salt for the biscuits. Cut in the cold butter using a pastry blender or your fingertips until the mixture resembles coarse crumbs.
Gradually add the milk, stirring with a fork until just combined. Do not overmix. Turn the dough out onto a lightly floured surface and gently fold it over a few times to create layers. Pat the dough into a 1 to 1.5-inch thick rectangle. Cut out 6-8 biscuits using a biscuit cutter or the rim of a glass.
In a large oven-safe Dutch oven or heavy-bottomed pot over medium heat, melt the butter for the filling. Add the shallots and sauté for 5 minutes until softened. Add the garlic, carrots, celery, and 1 teaspoon of salt; cook for another 5 minutes, stirring occasionally.
Sprinkle the flour over the vegetables and cook for 1-2 minutes, stirring constantly. If using wine, add it now and let it bubble and reduce slightly. Gradually whisk in the chicken broth, milk, and cream until smooth. Bring the mixture to a simmer and cook until it thickens into a creamy sauce, about 5-7 minutes.
Stir in the shredded chicken and frozen peas. Season the filling with the remaining 1 teaspoon of salt, pepper to taste, and a tiny squeeze of lemon juice. Remove from heat.
Arrange the prepared biscuits on top of the chicken filling. For extra browning, you can brush the tops of the biscuits with a little milk or melted butter.
Bake for 15-20 minutes, or until the biscuits are golden brown and cooked through, and the filling is hot and bubbly. Let the pot pie rest for 10 minutes before serving.
Notes
This recipe can be made vegetarian by omitting the chicken and using vegetable broth instead of chicken broth. For a shortcut, canned biscuits can be used in place of homemade.