Add the dry spaghetti to a large, wide pan or deep skillet, gently bending the strands so they fit in an even layer across the bottom.

Pour the vegetable broth and whole milk into the pan, making sure most of the spaghetti is submerged in the liquid.

Sprinkle in the fine sea salt and freshly ground black pepper, then add the minced garlic to the pan, distributing it as evenly as you can.

Place the pan over medium-high heat and bring the liquid to a gentle boil, using tongs to nudge and separate the spaghetti so it does not clump together.

Once boiling, reduce the heat to medium and cook the spaghetti, stirring and turning it frequently, until it is al dente and most of the liquid has been absorbed, about 10 to 12 minutes.

Lower the heat to low, then add the unsalted butter to the pan and stir until the butter has fully melted and coats the pasta.

Gradually sprinkle in the finely grated Parmesan cheese, stirring constantly, until it melts into the remaining liquid and forms a creamy sauce that clings to the spaghetti.

Taste the pasta and adjust the seasoning with a little more fine sea salt or freshly ground black pepper if needed, then remove the pan from the heat and let the pasta sit for 1 to 2 minutes so the sauce can thicken slightly before serving.