In a medium sized frying pan melt the butter over medium heat. Pulse the onion in a food processor until finely chopped and add to the melting butter along with the minced garlic. Cook until the onions are translucent, about 5 minutes.
While the onion is cooking pulse the ham in the food processor until finely ground. Set aside.
Sprinkle the flour over the cooking onion mixture, mix until all flower is moistened by the butter and cook, slightly browning the flour mixture, about 3 minutes.
Slowly add the milk, a little at a time, mixing constantly until all of the flour mixture is evenly combined. Repeat milk additions and mixing (at least 4 times) until all milk is added and sauce begins to thicken.
Add in the ham to the paste like mixture and stir well to combine, removing from heat. Allow mixture to cool in the refrigerator until it is firm and cool enough to handle.
Breading
Take the filling out of the fridge and roll into small logs, about 2-3 inches long and 1 inch wide.
Whisk the egg and milk in a shallow bowl and place the bread crumbs on a plate to set up your breading stations.
Dip each rolled croquette into the egg mixture and then roll in the bread crumbs.
Place the breaded croquettes into an air fryer basket and coat with cooking spray. Flip croquettes to coat all sides with the spray. Cook in an air fryer at 400F for 5 minutes. Serve with lime and enjoy!