In a blender, puree 1 cup sweet peppers, 1 cup tomatoes, 15 sea urchin fillets, the whole cloves garlic, 1 to 1 1/2 cups extra-virgin olive oil and lemon juice to taste. Season with salt and white pepper.
Bring a pot of salted water to a boil. At the same time, in a hot pan, add the chile de arbol, the minced garlic, the remaining 1 cup roasted peppers and roasted tomatoes along with some olive oil. Saute over high heat until softened.
Drop the pasta. When the pasta is ready, remove and add to the tomatoes and peppers. Quickly add the sea urchin puree and a little pasta water. Toss the pasta with the sea urchin. Check the seasoning. It should coat the pasta and slightly resemble carbonara. Plate the pasta, finish with extra-virgin olive oil, chopped chives and breadcrumbs.
Whisk the eggs, egg yolks, oil and water together. Mound the flour on a work surface and make a well in the middle. Add the salt. Slowly add the wet ingredients. Knead until the ingredients are incorporated and the dough is soft and pliable, 3 to 4 minutes. Add more water and/or oil if the dough is dry and crumbly.
Roll the dough with a pasta machine on the second to lowest setting into sheets. Cut with a knife into strands 1 1/2 inches wide by 6-inches long.