In a skillet over medium heat, add butter and oil. When it starts sizzling, add finely chopped white onions and sauté them for 1 minute. Add pressed garlic and sauté for 1 minute.
Add flour, stir until the ingredients are coated and cook for 1 ½ to 2 minutes.
Pour in half-and-half, and season with paprika, rosemary, oregano, thyme and nutmeg. Stir very well until the texture gets smoother and the sauce thickens
Add ham and parsley; blend into the mixture.
Remove from the heat and transfer the mixture to a bowl. Cool for 10 minutes before covering and transferring to the fridge for 5 hours.
In a shallow dish, add breadcrumbs and season with freshly ground black pepper; mix and set aside.
In another shallow dish, add an egg and beat it before adding milk. Whisk until blended and set aside.
Take a small amount of ham mixture, roll it into a ball and then form a cylinder with it by gently spreading it out between your hands.
When they’re all done, working with one at a time, dredge in the breadcrumb mixture making sure it’s entirely covered, shaking off any excess. Place in the egg mixture and ensure it’s well coated. Return the croquette to the breadcrumbs and evenly coat again.
Transfer to a plate while doing the others.
In a shallow skillet, bring the oil temperature to 375ºF.
Working in batches, carefully place croquettes in the oil and fry them until golden brown, about 3 minutes.
Using a slotted spoon, transfer them to a plate lined with paper towels to absorb any excess oil. Set them aside while doing the remaining ones. Makes 10 croquettes.