Place the garlic, pine nuts, olive oil, ice cubes, basil, and a pinch of sea salt in a blender. Blend for several minutes or until the pesto reaches a creamy, smooth consistency. Scrape down the sides and blend for another 30 seconds.
In the meantime, bring a large pot of water to a boil. Generously salt with sea salt and cook your pasta until al dente. Pour the pesto into a mixing bowl that is large enough to accommodate the pasta as well. Gently fold in the Parmigiano Reggiano, Pecorino Romano, and softened butter (if using). Taste and season with more sea salt if necessary.
Add the hot pasta directly into the mixing bowl. Stir to coat. Slowly add a little bit of pasta water until you get a creamy sauce coating the pasta. Pour into individual serving bowls.