Make carrot mash: In a medium saucepan, combine carrots, chopped apple, garlic, bay leaf, and broth. Bring to a simmer, cover, and cook until carrots and apple are very tender, 10 to 15 minutes. Discard bay leaf and transfer mixture to a high-speed blender or food processor. Add butter and process until smooth, scraping down sides of bowl as needed. Taste and season with salt and pepper. Cover to keep warm.
Preheat oven to 425 degrees F and line a rimmed baking sheet with parchment paper. In a bowl, toss together apple wedges, 2 teaspoons oil, sage leaves, and a pinch of salt. Spread on baking sheet and roast, turning once, until apple wedges are tender, about 10 to 12 minutes. Let cool slightly.
In a small bowl, whisk together broth, mustard, and maple syrup. Pat pork medallions dry; press with the side of a chef’s knife to ¼-inch thickness. Season all over with salt and pepper. Warm 1 tablespoon oil in a large skillet over high heat. Add pork and cook until seared, turning once, about 1 to 2 minutes per side. Transfer to a plate; cover.
Reduce heat to medium. Add onion to skillet; cook, stirring occasionally, until tender and beginning to caramelize, about 5 to 7 minutes. Pour in broth mixture and stir to release any browned bits in skillet. Bring to a boil. Reduce heat, add chopped sage and simmer until liquid has reduced by about one-third, 6 to 8 minutes.
Stir in arrowroot mixture. Add pork and any accumulated juices from plate to skillet along with apples and turn to coat pork with sauce. Cook until pork is cooked through and sauce thickens, about 2 to 4 minutes longer. Divide mash among 4 plates, top with pork, apples and sauce, and serve.