Heat the oil and butter in a medium saucepan over medium heat until butter is melted. Add the flour and cook 1-2 minutes while whisking frequently. Gradually add the milk while whisking and continue to cook another 2-3 minutes. The mixture should be smooth. Switch to a wooden spoon and stir in the ham, cheese and nutmeg. Cook another 1-2 minutes while stirring. The mixture will pull away from the sides of the pan. Taste the mixture and add Kosher salt if desired. If the ham and cheese are salty so you probably won’t need to add any additional salt.
Transfer the mixture to an 8x8-inch baking tray and spread it out so that it is even. Let the mixture cool, then cover it with plastic wrap and refrigerate at least 2 hours or overnight.
When ready to cook the croquettes, lightly beat the eggs in a shallow dish. Mix the panko and 1/2 teaspoon Kosher salt in another dish. Scoop up tablespoons of the cooled filling and form them into balls. Dip each ball into the egg and then the panko. Place the completed croquettes on a wire rack or baking sheet and refrigerate for 20 minutes. The croquettes must be chilled before frying otherwise they may fall apart in the oil.
Meanwhile, make the aioli by pureeing the mayonnaise, garlic, lemon juice and smoked paprika in a blender or mini food processor. Transfer to a small bowl and refrigerate until ready to serve.
Pour enough vegetable oil into a large stockpot to reach a depth of 1 inch and heat over medium high heat. Working in batches, fry the croquettes in the oil, turning them on all sides, until golden brown and crispy, about 2 minutes. Drain on paper towels. Serve immediately with the smoked paprika aioli.