Preheat oven to 450 degrees. In a food processor, chop carrots, celery, onion, and panko with egg white until smooth, about 1 minute. Transfer to a large bowl; mix in beef, 1 teaspoon salt, and 1/8 teaspoon pepper.
On a foil-lined baking sheet, form meat mixture into a 6-inch-long loaf; brush with half the barbecue sauce. Bake until an instant-read thermometer inserted in loaf registers 160 degrees, 25 to 30 minutes, brushing halfway through with remaining sauce.
Meanwhile, in a medium saucepan, cover potatoes with cold salted water by 1 inch. Bring to a boil; reduce to a simmer, and cook until potatoes are easily pierced with a knife, 15 to 20 minutes. Drain; return to pan. Stir over medium until a starchy film forms on bottom of pan, about 1 minute. Remove from heat; mash with buttermilk until smooth. Season with salt and pepper.
While potatoes cook, set a steamer basket in a saucepan filled with one inch water; bring to a gentle boil. Add green beans; cover, and cook until crisp-tender, 4 to 6 minutes. Transfer to a bowl; add oil, season with salt and pepper, and toss. Serve meatloaf with mashed potatoes and green beans.