How to Make It
Prick potatoes all over with a fork; microwave at high 5 minutes. Turn and cook until potatoes can be easily pierced with a fork all the way to the center, 5 to 7 more minutes. Carefully transfer to a cutting board to cool.
Bring a medium pot of water to a boil. Add carrot; cover and cook until tender, 10 to 12 minutes. Drain and transfer to a large bowl. Scoop flesh from cooked potatoes; add to carrots (discard potato skins). Mash with a potato masher.
Heat oil in a large skillet over medium. Add onion, parsley, and garlic; cook until soft, about 4 minutes. Stir in ham. Remove from heat; add to potato mixture. Stir in cheese, pepper, and salt.
Divide ham mixture into 18 portions (about 3 tablespoons each); roll each portion into a log. Place flour on a large plate; roll logs in flour, pressing to flatten slightly. Shake off excess flour; freeze for 30 minutes.
Preheat oven to 450°F. Coat a large rimmed baking sheet with cooking spray. Dip logs in egg wash; dredge in breadcrumbs. Transfer logs to prepared pan. Spritz generously with cooking spray. Bake at 450°F until golden and heated through, 20 to 23 minutes.