Croquettes are adorned with parsley for these bite-size appetizers.
Heat butter and oil in a medium saucepan over medium heat until butter has melted. Add onion; season with salt and pepper. Cook, stirring occasionally, until onion is translucent, about 3 minutes. Add flour; cook, stirring, 1 minute. Whisk in milk, and cook, whisking, 3 minutes. Whisk in ham and 1/3 cup cheese. Season with salt and pepper as desired. Spread mixture onto a baking sheet, and let cool completely. Mixture can be refrigerated in an airtight container up to 2 days.
Whisk together eggs in a shallow dish. Stir together breadcrumbs and remaining 1/4 cup cheese in another shallow dish. Scoop tablespoons of cooled mixture, and shape with floured hands into 2-inch ovals. Working with 1 oval at a time, coat in beaten egg, then in breadcrumb mixture. Transfer to a baking sheet lined with parchment paper.
Heat 2 1/2 inches oil in a large, heavy stockpot until it registers 375 degrees on a deep-fry thermometer. Working in batches to avoid crowding, fry croquettes, flipping once, until dark golden brown, 1 to 2 minutes total. Using a slotted spoon, transfer to paper towels to drain. (Adjust heat between batches as needed to keep oil at a steady temperature.) Serve warm, garnished with parsley sprigs.