Slice the eggplants into batons about 3 inches long and 1 inch thick. Place them in a large bowl, sprinkle with coarse salt, and submerge in room-temperature water for at least 10 minutes.

Whisk together the soy sauce, vinegar, wine, cornstarch, sugar, and broth in a small bowl until the starch is completely dissolved.

Place the TVP in a strainer and pour boiling water over it until submerged; let it soak for 5 minutes to rehydrate. Meanwhile, boil the wheat noodles until very chewy, about 2 minutes less than the package directions. Drain the noodles, rinse under cold water, and toss with the small amount of oil to prevent sticking.

Heat the larger amount of oil in a skillet or wok over medium-high heat until shimmering. Add the onion, garlic, and ginger and sauté until aromatic, about 2 minutes.

Add the bell pepper and bean paste to the pan. Increase the heat to high and cook for 2 minutes, or until the pepper is just tender.

Drain the eggplants well and add them to the pan. Cook over medium-high heat for 6 minutes, stirring occasionally, until the eggplant is tender.

Give the sauce a final stir to redistribute the starch, then lower the heat to medium and pour it into the pan. Stir constantly as it simmers for 2 minutes, until the sauce thickens and coats the vegetables. Season with salt as needed.

Divide the noodles into bowls and top with the hot eggplant mixture. Mix the sauce into the noodles and serve immediately while hot and saucy.