To prepare Mushroom Bolognese:
Place dried mushrooms in a bowl and cover with boiling water; let sit for 15 minutes. Meanwhile, pulse carrots, parsnips, celery, shallots, bell pepper and mushrooms together in a food processor until finely chopped. This may need to be done in batches.
Heat oil in a 12-inch heavy skillet over moderately high heat until hot but not smoking. Add chopped vegetables, rosemary and a generous pinch of salt and pepper and sauté, stirring occasionally, until vegetables are golden brown and tender, about 12 minutes.
Lift mushrooms out of soaking liquid, squeeze excess liquid back into bowl (reserve 2 cups soaking liquid) and rinse mushrooms well to remove any grit. Finely chop mushrooms, then add to vegetables in skillet along with tomato paste. Cook over moderate heat, stirring, 1 minute. Add wine, milk and reserved mushroom-soaking liquid and bring to a boil. Reduce to a gentle simmer and cook, stirring occasionally, until the liquid has reduced by half, about 30 minutes. Season well with salt and pepper. Sauce can be made up to 2 days ahead, or frozen for up to 2 months. Cover and chill until ready to use. Warm gently in a saucepan before assembling lasagna.
Cook the Lasagne Pasta:
Follow the package instructions for par-boiling lasagna pasta, or purchase no-boil pasta. Set cooked pasta aside until ready to assemble.
To Assemble the Lasagne:
Combine cheeses, parsley and sage in a bowl. Spread ½ cup mushroom sauce on the bottom of the baking dish and top with a single layer of noodles, ¾ cup mushroom sauce and a generous handful of cheese. For the next layer, top with ¾ cup mushroom sauce, a generous handful of cheese and a layer of squash slices. Repeat the layering process 4 more times, alternating the pasta layer with a layer of squash. Top with the remaining cheese.
Cover loosely with foil and bake until bubbling and squash is tender when pierced with a knife, about 45 minutes. Remove foil and continue to bake until cheese begins to brown, 10—15 minutes. Let stand 15 minutes before slicing.