Preheat oven to 375 degrees F.
Cut squash in 1/2 and scrape out seeds. Spread 1 tablespoon molasses in the cavity. Season with salt and pepper. Place cut side down on a roasting pan. Cook in the oven until very soft, about 1 hour. (See Chef's Notes.)
Let cool to room temperature and scoop out flesh into the work bowl of a food processor.
Puree squash until smooth, then spread on a baking sheet and return to the 375 degree oven to dry, about 10 minutes. The consistency will be like mashed potatoes. Scrape into a large mixing bowl.
Heat the butter in a saucepan over medium-low heat until it begins to brown. Immediately remove from heat and add remaining 1 tablespoon molasses and all the vinegar. Add to squash with mascarpone, Parmesan, cinnamon, and nutmeg. Season, to taste, with salt and pepper and mix well. The recipe can be made ahead to this point (makes 2 cups filling,) Cover well and refrigerate 4 hours or up to 2 days.
To fill the raviolis: Lay out a sheet of pasta dough on a lightly floured board. Cut into circles with a 3 1/2 inch pastry cutter. Put 1 tablespoon squash filling in the center of 1/2 the rounds using either a pastry bag or a small spoon. Leave a 1/2-inch border all around the filling. Moisten borders with water and top with remaining rounds of dough. Press all the air out and seal firmly by pressing all around with fingertips. Lay raviolis out to dry on a lightly floured board or baking sheet and lightly flour the tops. Repeat until you run out of dough and/or filling. To cook, boil in lightly salted water until tender, about 3 minutes. Reserve 2 ounces of the cooking water.
Uncooked, filled raviolis may be used immediately or frozen for 2 months. Lay them out on sheet pans and place in freezer until frozen. Transfer to plastic bag.
For the Sage Brown Butter: While raviolis are cooking, in a large saute pan, melt the butter with the sage and a pinch of salt until it foams and becomes light brown. Reserve.
On medium heat toss the cooked raviolis in the sage butter then transfer to a serving platter or dishes. Add 2 ounces of the cooking water to the pan and swirl with any residual butter. Spoon the butter sauce over the raviolis, then finish with a generous grating of Parmesan and bittersweet chocolate.