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Preheat the oven to 350°F and grease a 9 x 13-inch casserole pan.
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Bring a large pot of well-salted water to a boil. Cook the lasagne noodles until al dente. Drain and reserve.
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While the pasta is cooking, cook the sausage and onion in a wide-bottomed skillet with the two tablespoons of olive oil until the onion is translucent and the sausage is fully cooked, about 10 minutes. Be sure to break up the sausage as much as possible as it cooks. Drain on a sheet pan lined with paper towels until slightly cooled.
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In a large mixing bowl, mix together the drained sausage and onion, the grated asiago cheese, half of the grated Parmigiano-Reggiano, and the roasted butternut squash.
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Take a nice stretch or pour yourself a glass of wine. You’re getting there! The hardest work is over.
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Spoon about a cup of béchamel onto the bottom of the greased casserole. Working one lasagne noodle at a time: Lay a lasagne noodle flat on a clean surface. (I use the sheet pan.) Spread a spoonful of béchamel on the noodle. Spread some of the sausage-squash-cheese filling over the noodle, then roll it up as tight as feels comfortable. Gently lay it down in the casserole. Repeat until all of the lasagna rolls are settled snugly in the casserole.
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Pour the remaining béchamel over the roll-ups and sprinkle with the remaining Parmigiano-Reggiano and a good few twists of black pepper.
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Bake, uncovered, for about 45 minutes, until bubbling. Let rest 10 minutes, tented with foil, before serving. Use an additional rosemary spring as a garnish if you like.
