Preheat your oven to 450° F. Prepare your 9-inch springform pan by lining both the bottom and the sides of the pan with parchment paper (I smeared a little vegan butter to both so the parchment paper would stick). Make sure the parchment paper along the sides is at least 1-inch taller than the pan.
Add extra virgin olive oil to a large skillet over medium-high heat. When the oil begins to shimmer (about 1 minute), add the Italian sausage. Cook until all sides are entirely browned. Remove the Italian sausage and set aside.
To the same pan, add a little more oil, followed by shallots, mushrooms, and rosemary. Let the mushrooms cook, unbothered, for about 3 minutes, before sprinkling with salt and pepper. Then, add the garlic. Toss the contents of the pan and allow the mushrooms to cook for another 3 minutes. Deglaze with red wine, using a wooden spoon to scrape up any brown bits. Reduce to medium heat and continue cooking until the wine has almost completely reduced (about 5 to 7 minutes). Place your mushroom mixture, together with the browned sausage, into a food processor. Pulse about 10 to 12 times, until you achieve a pebble-like consistency.
To a large and deep pan over medium-low heat, add the vegan butter. Once it has melted (about 1 minute), add the flour. Using a wooden spoon, stir it until it turns into a yellow paste. Then, pour in plant milk while using a whisk to stir it in. Doing so will evenly incorporate the roux so that you don't get a lumpy cream sauce. Once your roux is completely incorporated, continue stirring until the mixture starts to bubble (about 30 seconds). Then, add the vegan cheese. Incorporate into the sauce until it's smooth and creamy. Remove from heat.
Cook pasta in a large pot of boiling water, seasoned with salt. Cook for approximately 30% of the cook time provided for on the package. Drain pasta and set aside for pie assembly. DO NOT TOSS IN OIL (if that's your normal MO). Doing so will make it harder to keep the rigatoni upright.
While your pasta is cooking, quickly whisk together your ricotta mixture by mixing together the ricotta, 1 teaspoon of salt, 1 teaspoon black pepper, chopped parsley, vegan parmesan, and 1/2 cup of the mushroom mixture. Place the mixture into a piping bag or zip lock bag. Do not snip the top until you're ready to pipe the mixture into the cooked rigatoni.
Begin by adding 1/3 of the béchamel to the bottom of your springform pan. Then, spoon half the remaining mushroom-sausage mixture on top of the sauce. Using a spoon, gently stir the two layers together so that the mushroom mixture becomes part of the sauce. Next, starting from the outer edge and working in, add the cooked rigatoni on top, keeping them upright. I ended up using every last rigatoni. It should not be an overly tight squeeze, as you'll want pockets for potato slices and the sauce to fall in as it bakes.Pipe the ricotta mixture into each of the rigatoni, all the way to the top. Then, weave in the potato slices between the noodles. After that, add the remaining mushroom mixture, together with another 1/3 of the béchamel. Again, using a spoon, mix gently until the mushroom mixture becomes part of the creamy sauce. Finally, top the entire pie with the remaining béchamel (I do this to create a clean top that will brown evenly). Place the pie in the center rack of the preheated oven. Bake for about 45 minutes (turning the pie at around the 35 minute mark, so that both sides get evenly baked), until the top is golden brown. Remove from the oven and let it cool for 30 minutes before carefully unlatching the springform pan and pulling away the parchment paper. Slice and enjoy!