Bring a large pot of water (about 6 qt.) to a boil for the pasta. Meanwhile, fill a wide, shallow pan with water and bring to a gentle simmer. Crack each egg into a saucer and gently slide it into the simmering water. Poach the eggs until the whites are firm but the yolks are still runny, 2 to 3 minutes. Remove each egg with a slotted spoon, drain any water, and set on a warm plate.
When the pasta water is boiling, add the salt and pasta. Stir frequently to keep the pasta from sticking. Use the cooking time on the package as a guide, but start checking 2 minutes sooner. When the pasta is al dente (soft but with some resistance left), drain it in a colander, reserving about 2/3 cup of the cooking water. Return the pasta to the empty pot and toss with 2 Tbs. of the cooking water, the butter, and the grated cheese. If it seems too dry, add a few more spoonfuls of water and toss again.
Portion the pasta into six warm serving bowls, top each with a poached egg and the caviar, and serve right away.