Cut eggplant into 1/4-inch slices lengthwise, then cut those into 2-inch pieces. Lightly salt the slices and let sit for 30 minutes—heat 2 tablespoons of oil in a wok or large cast-iron skillet over high heat. Add half of the eggplant and cook, occasionally stirring, until tender and browned in parts, 4 to 5 minutes. Remove to a large bowl and cover to keep the eggplant warm. Repeat with 2 more tablespoons of oil and the remaining eggplant. Add 1 tablespoon of oil to the pan and the mushrooms. Cook, often stirring, until mushrooms are brown and soft, 4 to 5 minutes. Remove to the bowl with the eggplant.

Mix the hoisin, tamari, and plum sauce together in a small bowl; set aside. Cook the Udon noodles according to package directions.

Heat the remaining oil in the skillet over high heat. Add the jalapenos and onion until starting to soften, 3 to 4 minutes. Add the garlic and ginger; cook often stirring for 30 seconds. Add the onion mixture and basil to the eggplant and mushrooms. Toss the udon noodles with enough sauce to coat. Toss the eggplant mushroom mixture with the rest of the sauce and serve over udon noodles.