Combine flour, salt, and pepper in a large resealable plastic bag. Add short ribs, one at a time. Seal bag; shake to coat with flour mixture. Remove from bag; shake off excess flour. Let coated ribs stand at room temperature for 30 minutes.

Heat 2 tablespoons butter in a large skillet over medium-high heat. Add ribs; cook for 10 to 12 minutes or until lightly browned on all sides, turning frequently. Set ribs aside.

Add potatoes, carrots, and broth to slow cooker. Place ribs on top. Cover and cook on low for 8 to 8½ hours or until ribs are fork-tender. Add bok choy and ½ cup kimchi to slow cooker. Cover and cook for 30 minutes more.

Transfer potatoes to a medium bowl. Add remaining 4 tablespoons butter and 1 tablespoon cooking liquid; mash with a potato masher until smooth.

To serve, arrange ribs, carrots, and bok choy on top of mashed potatoes. Drizzle with cooking liquid, if desired. Serve with additional kimchi, if desired.