Cook the spaghetti according to package instructions until al dente, then drain and set aside.
Peel and chop the onion and garlic finely.
Remove cauliflower leaves and trim the stem, then cut into florets; discard thick, fibrous trunk pieces.
Pulse the raw cauliflower florets in a food processor until they reach a minced, rice-like texture; transfer to a bowl.
Pulse the walnuts in the food processor until coarsely minced (do not over-process into a paste).
Heat the olive oil in a non-stick frying pan over medium heat, then sauté the chopped onion and garlic for 2–3 minutes until softened and translucent.
Add the minced cauliflower to the pan and cook for 3–4 minutes, stirring occasionally.
Add the minced walnuts and cook another 3–4 minutes, stirring occasionally, until slightly toasted.
Stir in the sieved tomato puree, sweet paprika, oregano, basil, salt, and black pepper; simmer the mixture for 4–5 minutes to combine flavors.
Optionally stir in 1 cup of homemade vegan alfredo sauce at the end to make the bolognese creamy, then heat through for 1–2 minutes.
Serve the sauce over the cooked spaghetti and sprinkle with vegan parmesan cheese if using.