Preheat oven to 375 F.
Heat oil in a large Dutch oven or other oven safe stew pot over medium high heat.
Pat the elk chunks dry, then season with salt and pepper. Sear the meat in the stew pot, working on at least 2 sides of the meat, about 5 minutes total. Set aside the browned meat. Remove all but about 2 tbs fat from the pot.
Lower the heat to medium, then add the onions and apples. Cook until the apples are softened and lightly browned, about 5 minutes. Sprinkle in the flour and garlic, then cook, whisking frequently, for another 3 to 5 minutes.
Pour in the white wine to deglaze the pan, scraping up any browned bits. Add the apple juice, sage, vinegar, and honey. Let cook for about a minute.
Return the elk to the pot, along with any juices that may have dripped. Add enough broth to cover everything, then bring to a simmer. Cover and transfer to the oven to braise for 1 hour, or until the elk is fork tender.