COOK BUTTERNUT SQUASH: Place squash in a 6-quart soup pot and add water, covering by a couple of inches. Stir in a teaspoon and a half each of kosher salt and fresh ground black pepper. Cover and bring to a boil. Reduce heat to simmer and cook until squash is very tender—about 15 minutes. Drain, return to pan and add the tablespoon of salted butter along with the grated Parmigiano Reggiano cheese. Mash to desired consistency, adding a drizzle or two of milk for a thinner mash. Taste and adjust seasonings if needed.
COOK THE PORK CHOPS: Meanwhile, preheat oven to 375 degrees and place rack at center. To a heavy (cast iron is good) skillet, add 1 tablespoon canola or peanut oil. Heat over medium-high flame. Meanwhile, pat chops dry with paper towels, brush both sides of chops with canola or peanut oil and season very well with salt and pepper. Add chops to hot skillet and cook until quite brown on one side–3 or 4 minutes. Turn and brown the other side. Place skillet in oven and roast until chops are at 145 degrees F on an instant read thermometer or just barely done, still juicy, and somewhat rosy. Begin testing after 5 minutes. Remove, cover loosely with foil, and let rest in pan to keep hot.
SAUTE THE MUSHROOMS AND SHALLOTS: In a 9-inch skillet, heat the reserved teaspoon of butter along with one tablespoon of olive oil over medium-high heat. Add the sliced mushrooms, shallots, and minced sage. Season with a generous pinch of pepper, but no salt. Let sit a minute before stirring, then cook until the shallots are tender and the mushrooms are browned, perhaps 5 minutes or so. Sprinkle with a little bit of salt once the mushrooms are done. Taste and adjust seasoning. Cover to keep warm if necessary.
PLATE AND SERVE: Spoon about a cup of the mashed butternut squash into each of 2 shallow bowls, add chops at the center, and top with a couple of tablespoons of the mushroom mixture. Place green vegetables or tomatoes to the side of the mashed squash, if using. Serve hot.