Heat 2 tablespoons of the butter in a large skillet over medium-high heat until the foam begins to subside. Add leeks, carrots, and celery and cook, stirring often, until vegetables are barely tender, about 5 to 7 minutes. Set aside.
Warm remaining 4 tablespoons butter and the garlic in a large sauce pan over medium-low heat. Once the butter has melted, add the flour all at once and whisk constantly for one minute.
Add chicken broth and whisk until smooth, scraping down the sides with a spatula if needed. Increase heat to medium and add the milk, bay leaf, and thyme sprigs, whisking to incorporate the milk.
Increase heat to medium-high and, whisking frequently, bring mixture to a simmer. Once the mixture begins to bubble, whisk constantly until it thickens, about one or two minutes more.
Stir in chicken (or turkey), sauteed vegetables, and sherry, cooking for an additional minute or two. Stir in the peas, parsley, salt, pepper, and lemon juice. Remove thyme sprigs and bay leaf. Set aside.
Preheat oven to 400º F while preparing buttermilk biscuits as directed.
Transfer filling to a 9 by 13-inch casserole dish or divide it between two 10-inch pie dishes.Top filling with biscuits and brush them with egg beaten with 2 tablespoons milk (if not baking immediately, refrigerate biscuits in a separate container to add right before baking).
Bake in the bottom third of the oven until the filling is bubbly and the biscuits are golden brown, 25 to 35 minutes. If the biscuits brown too quickly, slip a baking sheet on the top shelf of the oven to diffuse the heat. Let sit for 10 minutes before serving.