Mix ground beef, chopped onion, celery, milk, chopped mushrooms, and panko together in a large bowl; shape into 10 patties.
Mix all-purpose and whole wheat flours together in a wide, shallow bowl. Dredge patties in the flour mixture just to coat and move to a platter, taking care to not stack the patties.
Heat 2 tablespoons oil in a large skillet over medium-high heat. Cook patties in hot oil until browned, 3 to 5 minutes per side. Arrange browned patties in a slow cooker crock.
Heat remaining oil in a small skillet over medium heat. Cook and stir sliced mushrooms and sliced onion in hot oil until tender, about 5 minutes.
Stir cooked mushroom and onion mixture with cream of mushroom soup, beef broth, Worcestershire sauce, onion gravy mix, and black pepper together in a bowl until smooth; pour over the beef patties in the slow cooker.
Cook on High until patties are very firm, hot, and grey in the center, 3 to 3 1/2 hours (or on Low for 4 to 5 hours). An instant-read thermometer inserted into the center should read 160 degrees F (70 degrees C).
