How to Make It
Preheat the oven to 375º. Place racks in the lower and middle thirds of the oven. Drizzle the garlic with olive oil, wrap it tightly in foil and roast on the bottom rack of the oven for 50 minutes. Lightly rub the potatoes with olive oil, prick them all over with a fork and bake on the lower rack for 45 minutes, until fork-tender. Line a large baking sheet with foil. Add the squash and rub with olive oil. Bake on the upper rack for about 30 minutes, stirring once, until soft.
Squeeze the roasted garlic cloves out of their skins into a small bowl and mash to a paste. Peel the hot potatoes and pass them through a ricer into a large bowl. Add the hot squash to the ricer and pass it into the bowl with the potatoes. Let cool slightly. Add the egg yolks, ricotta, parsley, 1 tablespoon of salt and 1 tablespoon of the mashed roasted garlic (reserve any extra for another use). Stir until combined. Sprinkle on the 1 1/4 cups of flour and gently stir it in. Scrape the dough onto a floured surface and knead gently until smooth but still slightly sticky.
Line a baking sheet with wax paper and dust with flour. Cut the gnocchi dough into 5 pieces and roll each piece into a 3/4-inch-thick rope. Cut the ropes into 1/2-inch pieces and transfer the gnocchi to the baking sheet.
Lightly oil another baking sheet. In a large, deep skillet of simmering salted water, cook half of the gnocchi until they rise to the surface, then simmer them for 1 to 2 minutes longer, until cooked through. Using a slotted spoon, transfer the gnocchi to the baking sheet. Repeat with the remaining gnocchi.
In a large nonstick skillet, melt the butter over moderate heat and cook until golden brown, about 2 minutes. Add the sage and thyme and cook for 20 seconds. Add the gnocchi and cook for 1 minute, tossing gently. Season with salt and serve, passing the cheese shavings at the table.