Cook the spaghetti in a large pot of boiling water for the time directed on its package. A strand should have the package’s intended doneness, with a firm bite rather than a mushy center.
Scoop out 1 cup of cooking water before draining, then place the drained spaghetti back in the pot. The reserved cloudy water will help loosen the sauce and blend in the butter.
During the pasta cooking time, heat 4 tablespoons butter in a skillet over medium-low, add the chopped garlic and a generous seasoning of salt, and stir occasionally for 1 to 3 minutes. Look for fragrant, softening garlic that remains pale, with no browned edges.
Mix the gochujang, honey, and vinegar into the garlic butter, then use medium-high heat to simmer the mixture for 3 to 4 minutes while stirring continuously. The sauce should become noticeably more concentrated as its sweet, tangy chili aroma develops.
Draw the spatula through the sauce along the skillet bottom and take the skillet off the heat once the cleared path remains open for about 3 seconds. A path that closes immediately indicates the sauce still needs more reduction.
Add the reduced sauce and the remaining 2 tablespoons butter to the spaghetti pot, then stir briskly until the butter melts. The strands should begin taking on an even, shiny coating as the butter blends through the sauce.
Work in small splashes of reserved cooking water as needed, tossing vigorously between additions, then taste and adjust with salt and pepper. Stop adding water when the noodles separate easily and the sauce clings without looking watery.
Finish with cilantro or scallions if using, and serve immediately. The noodles should reach the bowl glossy and evenly coated, without a separate layer of oil.