Boil the Noodles: Bring a large pot of heavily salted water to a rapid boil. Add the dry angel hair pasta and cook for roughly 4 minutes until just barely softened. Drain well and set aside.
Bloom the Garlic: Place a wide, heavy-bottomed skillet over medium heat. Add the cold butter. Once melted and lightly foaming, drop in the minced garlic and stir constantly for exactly 60 seconds until highly fragrant. Do not let the garlic turn brown.
Build the Roux: Sprinkle the dry all-purpose flour directly over the bubbling garlic butter. Whisk the paste vigorously for 2 solid minutes until it turns a light, sandy blonde color and smells lightly toasted.
Create the Liquid Base: Slowly stream the warm chicken broth and warm milk directly into the hot skillet. Whisk fiercely and continuously to dissolve any stubborn flour clumps and create a smooth liquid. Add the kosher salt, black pepper, and dried Italian herbs.
Thicken the Sauce: Let the milky liquid come to a very gentle simmer over medium heat. Let it bubble undisturbed for 5 minutes, occasionally scraping the bottom, until it heavily coats the back of a wooden spoon.
Melt the Cheese: Turn the stove heat completely off. Sprinkle the freshly grated parmesan cheese directly into the thickened gravy. Whisk violently until the cheese melts entirely, leaving a perfectly smooth, glossy sauce.
Coat and Serve: Dump the hot, drained noodles directly into the heavy cheese sauce. Toss repeatedly with kitchen tongs until every single strand is heavily smothered. Serve immediately while piping hot!