To prepare the chicken: Place the chicken thighs and salt in a deep saucepan., Add the water; it should cover the meat (if it doesn't, add more). Bring the water to a boil, reduce the heat to low, and cover., Simmer the meat for 10 minutes, then turn the heat off (leave the cover on), and let it rest for 35 minutes, until it's cooked through. This gentle method of cooking the meat will keep it nice and moist., Remove the meat from the pot, and spread it on a platter to cool., Preheat the oven to 450°F. Lightly grease a 2- to 3-quart casserole dish, 9" x 13" baking pan, or equivalent. While the oven is preheating, prepare the filling., To make the filling: Heat the butter over medium heat until it's melted. Add the flour and salt, and stir to combine., Gradually pour in the 2 1/2 cups broth, whisking constantly. Cook and stir the sauce over medium heat until it comes to a boil, then reduce the heat and simmer it for 5 minutes., While the sauce is simmering, discard any bits of skin or fat from the chicken, and tear the meat into 1" pieces., When the sauce is ready, add 1 1/4 cups of the milk or half-and-half, the meat, vegetables, salt, pepper, thyme, and sherry. Stir to combine., Understanding that the sauce will thicken a bit as the pie cooks, add additional milk or half-and-half, if desired, until it's the thickness you like., Bring the mixture to a simmer, then spoon the filling into the prepared dish. Place the dish in the oven, and let it cook as you prepare the biscuits., To prepare the biscuits: In a medium-sized bowl, whisk together the flour, baking powder, and salt., Work the butter into the dry ingredients until the mixture is crumbly, using a pastry fork or pastry blender, your fingers, or a mixer., Add 1 1/3 cups of the milk all at once, stirring quickly to make a cohesive dough; if the dough seems too dry, add the remaining milk., Turn the dough out onto a well-floured work surface and knead it a few times to smooth it out., Roll and pat the dough about 3/4" thick. Cut it into about 10 circles, squares, or whatever shape you like., To assemble and bake the pie: Remove the pan from the oven and place the biscuits atop the meat and vegetables, gently pressing them down a bit., Bake the pie for 15 to 18 minutes, until the biscuits are golden brown and the filling is bubbly. Check to make sure the biscuits are done inside. If they're not (or if the filling doesn't seem hot enough), tent the pan with aluminum foil and bake for an additional 10 to 15 minutes, until the biscuits are cooked through, and the filling is hot., Remove the pie from the oven, and allow it to cool for 10 minutes before serving., Storage information: Store any leftover pie, covered, in the refrigerator for up to five days; or freeze for up to three months.