Cut and salt the eggplant: trim the stems and cut the eggplant into batons about 3 inches long and 3/4 inch thick, or into thick half-moons. Toss with the kosher salt in a colander and let stand 10 minutes. Pat very dry with a clean towel or paper towels.
Mix the sauce: in a small bowl, stir the fish sauce, soy sauce, oyster sauce, sugar, and water until the sugar dissolves. Set it by the stove with the garlic, chilies, shallot, and basil all prepped.
Sear the eggplant in batches: heat a wok or large heavy skillet over high heat until very hot. Add 2 tablespoons of the oil and swirl. Lay in half the eggplant in a single layer and leave undisturbed for 2 minutes. Toss and cook 2 to 3 minutes more until golden at the edges and tender. Move to a plate. Add the remaining tablespoon of oil and repeat with the second batch.
Bloom the aromatics: return the wok to medium-high heat. Add the garlic, chilies, and shallot. Stir-fry 30 to 45 seconds until the garlic is fragrant and just pale gold at the edges.
Sauce and toss: return all the eggplant to the wok. Pour the sauce around the edges of the pan, then toss for 60 to 90 seconds until the eggplant is glossy and the sauce has reduced to a light coating.
Basil off the heat: take the wok off the heat. Fold in the Thai basil leaves and toss just until they wilt, about 20 seconds. Serve at once over jasmine rice with lime wedges on the side.