1. Heat oven to 200 degrees . In a large skillet, heat 1 tbsp of the oil over medium-high heat. Add mushrooms and cook 8 minutes, stirring a couple of times, until lightly browned. Add sliced shallots; cook 2 minutes. Mix in sage; remove to a bowl. Turn off heat and reserve pan.
2. Carefully split pork chops in half lengthwise (knife parallel to cutting board), making 8 thin chops. Pound each to 1/8-inch thickness on a cutting board in between 2 pieces of plastic wrap. On a plate, combine flour, 1/2 tsp of the salt and 1/4 tsp of the pepper. Dredge chops in flour mixture, gently shaking to remove any excess; place on another plate.
3. In same skillet, over medium-high heat, add 1 tbsp of the oil. Saute 4 of the chops for 2 minutes; flip and cook another 2 minutes. Remove to a baking sheet and place in warm oven. Add remaining 1 tbsp oil to skillet and repeat with remaining 4 chops.
4. Pour chicken broth into empty skillet. Whisk in mustard and release brown bits on bottom of pan. Bring to a simmer. Reduce heat to medium and simmer 3 to 4 minutes, until slightly thickened. Stir in mushroom mixture.
5. Meanwhile, reheat squash in a medium, lidded pot with milk, remaining 1/2 tsp salt and remaining 1/4 tsp pepper. Serve pork over squash. Ladle mushroom sauce on top.