Mix gochujang, miso, and 1½ cups warm water in a small bowl until blended. Set aside.
Boil pasta in salted water on high until al dente. Drain.
Heat 3 tbsp. oil in a large skillet over medium-high (I do not recommend cast iron for this, as eggplant will stick heavily).
Add eggplant and sear, drizzling with another 1 to 2 tbsp. oil.
Season eggplant lightly with salt and cook, undisturbed, until golden brown for several minutes.
Turn eggplant over and continue to cook until most of the eggplant are golden and becoming tender, about 5 minutes. Transfer to a plate.
In same pan, reduce heat to medium and add remaining 1 tbsp. oil, garlic, and ginger to skillet, stirring often until fragrant, about 1 minute.
Add tomato paste, stir, and cook until slightly darkened, about 1 minute.
Stir in gochujang mixture, combining well, and return eggplant to skillet.
Cook, stirring occasionally, until eggplant is very tender, about 6 minutes.
To gochujang mixture add noodles and butter, seasoning with salt as needed.
Cook until sauce is thickened, about 2 minutes. Remove to plate or individual bowls.
Scatter noodles with scallions, coconut, and peanuts.