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Make the Parm Broth: Cut 2 whole heads of garlic in half crosswise. In a medium pot, combine the garlic (skins and all), 1 ½ teaspoons black peppercorns, and ½ pound Parmigiano Reggiano rinds. Cover with 9 cups of water. Bring to a simmer over medium-high heat. Once simmering, reduce to a bare simmer and cook, for at least 1 hour, or up to 2. Stir in 1 tablespoon salt. Taste and add a bit more if needed.
Toast the pasta: In a medium pot, heat 2 tablespoons of olive oil over medium-high heat. Add ½ cup tiniest pasta you have, a big pinch of salt and a bunch of freshly ground black pepper. Cook, stirring the pasta, until it is deeply toasted and nutty, 2-3 minutes. Strain the parm broth right into the pot. Discard solids. Bring the broth to a simmer, and cook, stirring occasionally, until the pasta is tender.
Poach the eggs and serve: Working one at a time, crack 2 eggs into the pot of broth. Cover and cook, at a bare simmer, until the whites are set and yolks are as runny or cooked as you like them (3-6 minutes). Carefully transfer an egg to each serving bowl along with some broth and a few scoops of pasta. Season the eggs with salt and pepper. Finely grate lots of fresh parm over each bowl and drizzle with more olive oil.