Soak bok choy in a large bowl of cold water, swishing to loosen any grit. Drain, cut in half lengthwise, and dry carefully with paper towels. Heat oil in a large, heavy-bottomed saucepan or Dutch oven over high heat until shimmering. Add halved bok choy, cut side down, working in batches if necessary to avoid crowding pan, and sear until deeply browned, about 6 minutes. Remove to a large plate or sheet pan.
Lower heat to medium and add ginger and garlic, stirring constantly, until fragrant, about 30 seconds. Add water and vinegar and bring to a simmer, stirring and scraping bottom of pan with a wooden spatula or spoon to loosen any browned bits. After 1 minute, add soy sauce and brown sugar. Add bok choy back to pan and cook, uncovered, until bok choy leaves are wilted and stems are crisp-tender, 6-8 minutes. Remove bok choy to serving platter.
Continue to cook liquid in pan over medium heat until it reduces to a glaze of desired consistency, 6 to 8 minutes. Spoon glaze over bok choy and garnish with sesame seeds and chives. Serve with hot cooked rice, if desired.