Melt 2 tablespoons of the butter in a large pot or deep skillet over medium heat. Add the garlic and cook for 1 to 2 minutes, until fragrant.
Add the chicken broth, half and half, and 1 teaspoon of the black pepper and stir to combine. Bring to a boil, then add the pasta and stir. Reduce the heat to a simmer and cook uncovered, stirring often, until the pasta is tender and the sauce has thickened up, about 15 minutes.
Add the parmesan and pecorino all at once, stirring until fully melted and the sauce is smooth and glossy.
Add in the remaining 1 tablespoon of butter and the remaining black pepper, stirring until the butter is melted and glossy and the black pepper is mixed all through. Taste and add more pepper to your liking.
Remove from heat and serve immediately.