Step 1Bring a large pot of water to a rolling boil with the kosher salt (about 1 tablespoon of salt for every 2 quarts of water). Add the spaghetti and cook, stirring occasionally, until it is al dente, 12 minutes, or according to the package directions.
Step 2While the spaghetti cooks, heat the chorizo with the butter and garlic in a large, deep skillet until the sausage starts to crisp and the garlic to brown, about 5 minutes. Remove and discard the garlic.
Step 3When the pasta is done, drain it in a colander and put it in a large bowl. Scrape the chorizo mixture in. Immediately add the eggs and half the cheese. Toss until all ingredients are well mixed and the pasta well coated. Season with salt and pepper to taste. Add the remaining cheese, mix well and serve at once.