Diced and sauteed fancy mushrooms such as shitake, porcini, or crimini are a great vegetarian substitute for the ham.
- In a large skillet, melt the butter over medium heat. Add the onions and cook, stirring, for about 2 minutes until they are translucent but not browned. Whisk in the flour and stir continuously to smooth. Continue stirring with a wooden spoon until the mixture thickens and forms a paste, another 3 to 5 minutes. Slowly add the milk, whisking constantly to incorporate and until it is smooth. Add the nutmeg and season lightly with salt and white pepper. (The ham is likely to be salty, so go easy on the salt. You can add more later if necessary.)
- Add the diced ham and stir to combine.
- Pour the mixture into a well-oiled medium-sized shallow bowl with straight sides. Let it cool to room temperature, then wrap in plastic and place in the refrigerator for at least 2 hours or overnight to fully chill and allow the mixture to become firm.
- Place 3 medium, shallow bowls, one beside the other. In the first, place 1/4 cup of flour. In the second, place the breadcrumbs. In the third, the beaten eggs. Keep them in this order for easy breading.
- Lightly flour your hands. Using a measuring spoon, scoop up 1 tablespoon of the mixture. Lightly roll it in flour, shaking off excess, then roll it gently in between your hands and shape it into a croqueta form. (You can do the traditional oval or a circle.)
- Dip the croqueta in the breadcrumbs, then in the egg, then in the breadcrumbs again. Place on a baking sheet. Repeat with the remaining mixture.
- Once the croquetas are all shaped, place them in the refrigerator for half an hour before you’re ready to cook them. (If you don’t want to cook them all at once, you can also freeze some and thaw when desired.)
- To fry, line a baking sheet with paper towels. Into a medium fry pan, pour olive oil so that it’s about one inch deep. Heat the oil to 375 degrees. Fry the croquetas several at a time, being careful not to overcrowd the pan. Lower the heat if the oil gets too hot or starts to smoke. Turn the croquetas once to cook on both sides. The whole process should take about 2 to 3 minutes. Drain them on the paper towel-lined baking sheet and serve immediately.