Preheat the oven to 350ºF.
Spread the carrot pieces and coins on a baking sheet in a single layer, drizzle with the olive oil, and toss to coat. Roast for 10 to 15 minutes, or until lightly browned but not crisp. Set aside, reserving the coins in a separate bowl.
Put the apple pieces into a bowl.
In a pan, bring the apple cider just to a boil, then pour it over the apple pieces and set aside for 3 minutes. Strain the apple pieces, reserve the cider, and add the apple pieces to the carrot pieces.
Add the wine to the reserved cider.
In a large pan, bring the stock to a boil, then reduce the heat to low.
Melt the butter in a soup pot over medium heat. Add the shallots and cook until translucent. Add the rice and stir to coat each grain with butter. Add the cider-wine mixture and cook, stirring frequently, until all of the liquid is absorbed. Add the warm stock, 1 cup at a time, stirring constantly, until the rice absorbs most of the liquid before adding more. Allow 25 to 30 minutes. Add the sage, carrot and apple pieces, cheese, and salt and pepper, to taste. Stir until the cheese melts.
Transfer to a serving bowl and garnish with carrot coins and sage leaves.