Working in 2-3 batches, add the mushrooms to the bowl of a food processor and pulse until finely chopped. Transfer to a large bowl and set aside.
Wipe out the food processor and repeat with the cauliflower florets, pulsing until the pieces are roughly the size of rice. Transfer to another bowl and set aside.
Heat ¼ cup of the olive oil in a Dutch oven or heavy pot over medium-high heat. Sauté the mushrooms until softened, stirring occasionally, 5-6 minutes. Stir in the onion and 2 TBSP more of the olive oil and cook until very soft, another 6-8 minutes.
Add the chile, garlic, and rosemary to the pot and continue cooking for another 3 minutes, until fragrant. Stir in the tomato paste and let darken, about 2 minutes, then add the wine and cook 3-4 minutes to burn off the alcohol. Next add the diced tomatoes, balsamic vinegar, tamari, prunes, fresh thyme, salt, and pepper. Stir to combine and let cook 5 minutes.
Pour the cauliflower into the sauce, toss well to coat, then cook 8-10 minutes, or until it starts to stick to the bottom of the pot. Season to taste, then cover and reduce the heat to low while you cook the pasta.
On a separate burner, bring a large pot of salted water to a boil, and cook the pasta 1 minute less than the package instructions, reserving 1 cup of starchy water. Transfer the cooked pasta and reserved pasta water to the bolognese and stir to coat the noodles. Increase heat to medium and cook for 2-3 minutes, or until the rigatoni is al dente.
Remove from the heat and stir in remaining 2 TBSP of olive oil, lemon juice, parsley, and grated Parmesan. Divide into bowls the serve, then top with more parmesan and parsley, a drizzle of olive oil, and a pinch of flaky sea salt.