Bring a large pot of water to a boil and salt it well. Peel the potatoes and cut them into medium-sized cubes. Boil the potatoes until cooked through (about 10 minutes).
Add the saffron to a small ramekin and pour about ½ tablespoon of hot water from the potatoes on top. Leave it to infuse while the potatoes are cooking.
When the potatoes are cooked, remove them from the water with a slotted spoon and transfer them to a bowl. Add the saffron and water, 1 tablespoon of grated Parmigiano Reggiano, 1 tablespoon of olive oil and a good pinch of salt and pepper.
Blitz everything to a super smooth puree using an immersion blender. Check for seasoning and add any salt if needed.
Add the pasta to the boiling water and cook until al dente.
Meanwhile, remove the sausages from their skins and fry them in a large pan, breaking the meat up with a wooden spoon. Once cooked, remove the sausages to a bowl and set aside.
Saute the onion until soft in the same pan until soft (add a little olive oil if needed).
When the pasta is almost done, add most of the sausage back to the pan (keep some for topping) along with all of the potato puree. Heat it on a medium low heat and add some pasta water a little at a time to loosen it.
Transfer the pasta to the sauce and stir until coated adding more pasta water to help loosen the sauce.
Serve in bowls topped with more Parmigiano Reggiano and leftover sausage.
Video
Notes
Emulsifying the sauce - because the sauce is made from a potato puree it's naturally very thick. Use pasta water to loosen the consistency of the sauce so it coats the pasta. You're looking for a carbonara-like consistency.
Using an immersion blender - I find this to be the easiest method for a super smooth consistency. You can use a blender instead just make sure to have a spatula ready to scrape out all of the potato.
Using up leftovers - if you have any leftover pasta you can heat it up with a little water to loosen the sauce.