Heat 1 tablespoon butter in a medium pan over medium heat. Sauté onions until translucent.
Add rest of butter, letting it melt. Add flour and cook for 3-4 minutes, stirring constantly until there are no lumps.
Gradually add milk, salt, nutmeg and freshly ground pepper to taste. Cook stirring constantly, until sauce is thickened and smooth.
Add minced ham (or chicken, hamburger) and stir. Remove from heat. Taste and adjust seasoning if necessary.
Place mixture in a sheet pan, cover with plastic wrap and allow to cool. Refrigerate at least 1 hour until mixture becomes firm. If you are preparing a day or so ahead, you can cover tightly and keep refrigerated until you are ready to fry croquettes.
Spread breadcrumbs on a flat surface. Beat eggs with water in a small, wide bowl. Cover your hands in flour, then divide mixture into 1-inch oblong shapes and set on a plate so that they are not touching.
Dip croquettes in beaten egg and coat with crumbs by rolling on flat surface. Cover with plastic wrap and let rest in refrigerator, covered, for at least 15 to 20 minutes.
Pour enough vegetable oil in a medium to large frying pan to cover a depth of about 3 to 4 inches. Heat to about 360°F. Fry croquettes quickly, turning several times, until golden-brown.
Remove croquettes with a slotted spoon and set on a paper towel to absorb excess oil. Serve immediately.