Melt the butter: In a large, deep skillet or pot, melt the butter over medium heat.
Bloom the garlic: Add the garlic and cook for 60-90 seconds, stirring constantly, until fragrant but not browned.
Add the pasta and liquid: Add the broken pasta, broth, milk, salt, and pepper. Stir well and bring to a gentle simmer.
Cook the pasta in the sauce: Reduce the heat to medium-low and simmer uncovered for 12-15 minutes, stirring often, until the pasta is al dente and most of the liquid has been absorbed.
Finish the sauce: Remove from heat and stir in the parmesan until melted and the sauce turns glossy and clings to the noodles.
Serve: Top with parsley, more parmesan, and red pepper flakes if using.
Notes
Cook the pasta right in the broth and milk instead of boiling it separately, the starch it releases is what thickens the sauce. Stir often so it doesn't stick to the bottom of the pot.