Preheat broiler to high. Place bell pepper half, skin side up, on an aluminum foil-lined baking sheet; flatten with hand. Broil in preheated oven until blackened, 8 to 10 minutes. Fold foil around pepper to seal tightly; let stand 10 minutes.

Meanwhile, grate shallot onto paper towels using small holes of a box grater. Transfer grated shallot to a large bowl. Finely shred half of cheese using the small holes of box grater; shred remaining cheese using large holes. Add shredded cheese to bowl with shallot. Stir in mayonnaise, kimchi, chives, and black pepper.

Unwrap foil around bell pepper; peel and discard skin. (Note: If you want to skip Step 4, finely chop pepper, and stir into cheese mixture with smoked paprika instead.)

Place an 8-inch square or 11- x 7-inch disposable foil pan upside down on a cutting board. Pierce 6 to 10 holes at 1 end of bottom of pan, near edge, using a knife. Turn pan right side up. Place dry wood chips in one layer over holes in pan. Place bell pepper on other side of pan; cover pan tightly with foil. Turn on vent hood. Set pan over a gas or electric burner so holes come in contact with heat. Turn burner on high; let pan sit on burner until smoke begins to seep from foil, about 30 seconds. Reduce heat to medium-low; smoke 4 minutes. Remove pan from burner; carefully uncover. (If needed, spritz wood with water to prevent burning.) Transfer pepper to a cutting board. Finely chop; stir into cheese mixture.

Taste for seasoning; add salt, if desired. Store in refrigerator up to 2 weeks.