In a large skillet over medium-high heat, heat 1 tablespoon vegetable oil. Add half of bok choy, arranging cut side down in a single layer, and cook, undisturbed, until golden brown, 3 to 4 minutes. Transfer to a plate. Repeat with 1 tablespoon vegetable oil and remaining bok choy.
Reduce heat to medium. In same skillet, heat remaining 1 tablespoon vegetable oil. Add garlic and ginger and cook, stirring, until fragrant, 30 to 60 seconds. Return bok choy to pan; season with ½ teaspoon salt and toss to combine.
Add soy sauce and 1 tablespoon water. Cover and steam until bok choy is just fork-tender, 2 to 4 minutes. Uncover and continue to cook, tossing frequently, until liquid is evaporated, about 30 seconds more; season with remaining ½ teaspoon salt, if needed.
Transfer bok choy to a platter. Drizzle with sesame oil.
