Preheat oven to 350 degrees. Set a steamer basket or colander in a large saucepan filled with 2 inches water; bring to a boil, then reduce to a simmer. Place broccoli florets and stems in basket, cover, and steam until bright green and just tender, 4 to 6 minutes. Remove basket from pan; let cool slightly.
Brush a 5-by-9-inch loaf pan with oil. In a food processor, pulse onion, carrot, and celery until finely chopped; transfer to a large bowl. Pulse broccoli in food processor until finely chopped. Transfer to bowl.
Add breadcrumbs, chicken, egg, 1 tablespoon mustard, Parmesan, dill, and salt to bowl. Season with pepper. Using two forks, mix until combined. Transfer mixture to prepared loaf pan.
Stir together ketchup, remaining 2 teaspoons mustard, and brown sugar in a small bowl until smooth; brush evenly on top of loaf. Set loaf pan on a rimmed baking sheet and bake until glaze is caramelized and a thermometer inserted in center of loaf registers 160 degrees, about 1 hour. Let cool 20 minutes before serving.