For the Beef: Preheat oven to 300 degrees F. Heat oil and butter in a large Dutch oven over high heat. Season the ribs with salt and pepper and cook on both sides for 5 minutes, or until golden brown. Add all of the wine to the pan and use a wooden spoon to scrape up any browned bits from the bottom, add all of the remaining ingredients, except the mushrooms, and season with salt and pepper. Cover with the lid, braise on the stove top or place in the oven and let braise for 2 1/2 to 3 hours or until very tender. Remove the meat to a plate and strain the cooking liquids into a bowl. Clean out the pan and add the strained liquid into the pan. Add the mushrooms and cook over high heat until reduced by 1/3. Return the meat to the pan and cook for 5 minutes to heat through.
For the Pasta Gratin: Place milk and garlic in a small saucepan and bring to a boil. Remove from heat and let steep for 10 minutes, then remove the garlic. Melt butter in a small saucepan, whisk in the flour and cook for 1 to 2 minutes over low heat. Gradually whisk in the milk and cream and cook over medium-high heat until thickened, season with salt and pepper and nutmeg, remove from heat and let cool for 5 minutes. Preheat oven to 450 degrees. Place cooked pasta in a buttered medium flameproof baking dish, add the sauce and stir to combine. Sprinkle the Gruyere over the top of the pasta then sprinkle the Parmesan over the Gruyere. Bake for 10 to 15 minutes. Turn on the broiler and broil, rotating the dish, until the cheese is golden, about 3 to 4 minutes. Let stand for 5 minutes. Garnish with chives.