Heat the oven to 425°F. Toss the fennel with 3 Tbs. of the oil and season with 1/2 tsp. salt and 1/4 tsp. pepper. Line a baking sheet with kitchen parchment or foil. Spread the fennel mixture over the baking sheet. Bake until the fennel is soft and caramelized, turning once, about 30 to 35 minutes. Remove from the oven and reduce the oven temperature to 350°F.
Meanwhile, heat the remaining 1 Tbs. oil in a medium nonstick skillet over medium-high heat. Crumble the sausage into the skillet and cook until browned, stirring occasionally, 5 to 7 minutes. Transfer the sausage to a large bowl with a slotted spoon. Reserve 2 Tbs. of the drippings. (If there are no drippings, substitute 2 Tbs. butter).
Heat the 2 Tbs. drippings or butter in the same skillet over medium heat. Add the apple and season with a little salt and pepper. Cook until the apple is tender and golden brown on both sides, 5 to 7 minutes. Remove, add to the sausage; mix in the roasted fennel.
In a large bowl, whisk together the sour cream, apple cider, eggs, egg yolks, sage, and the remaining 3/4 tsp. salt and 1/4 tsp. pepper/ Fold in the bread and the sausage mixture. Let sit for 30 minutes, stirring occasionally, to saturate the bread.
Brush a 2-qt. (or 9x13 inch) baking dish evenly with the remaining 2 tsp. butter. Pour the bread mixture into the baking dish, distributing the ingredients evenly. Bake in the hot oven until lightly browned and firm, 40 to 45 minutes.
A roasted fennel, sausage, and apple bread pudding has all the ingredients to make a complete main dish. Serve it with a mixed green salad.