Bring a large pot of water to a boil. Toss the pork, 1 tablespoon vegetable oil, 1 teaspoon cornstarch, 1/2 teaspoon salt and a few grinds of pepper in a medium bowl. Whisk 3/4 cup water, the soy sauce, hoisin sauce, brown sugar, vinegar, 1/4 teaspoon salt and the remaining 2 teaspoons cornstarch in a measuring cup; set aside.
Add the noodles to the boiling water and cook as the label directs. Drain; keep the noodles in the colander. Meanwhile, heat 1 tablespoon vegetable oil in a large nonstick skillet over medium-high heat. Add the eggplant in a single layer and season with salt. Cook until browned and starting to soften, about 3 minutes per side. Remove to a bowl.
Increase the heat to high and add 1 tablespoon vegetable oil to the skillet. Add the pork in a single layer; cook, undisturbed, until browned on the bottom and only a little pink remains, 2 to 3 minutes (the pork will not be cooked through). Remove to a plate.
Add the remaining 1 tablespoon vegetable oil to the skillet. Add the bell pepper, ginger, eggplant and a pinch of salt. Cook, stirring, until the bell pepper starts softening, about 2 minutes. Add the soy-hoisin mixture and bring to a simmer. Return the pork to the skillet and cook through, about 30 seconds.
Run the noodles under warm water to loosen and shake well to drain. Divide among bowls and top with the stir-fry.