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Heat the butter and olive oil in a saucepan or skillet over medium heat. Once the butter melts, add the onion and season with salt and pepper.
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Once the onion is translucent, add the flour and stir, letting the flour cook for 1 minute.
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Next, whisk in the milk. Continue whisking until the flour is not longer clumpy. Continue to whisk, letting the mixture cook for 3 minutes, it will get very thick.
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Add 1/3 cup of cheese and the chopped ham, and stir to combine. Let this mixture cool in the fridge until it is cool enough to handle. To form the croquettes, scoop up a tablespoon of the mixture and then form into an oval. Continue using all of the dough and making about 16 croquettes.
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Crack the eggs into a bowl and whisk. Combine the panko, breadcrumbs, and remaining cheese in another bowl. One by one, roll the croquettes in the breadcrumbs, then the egg, and then back in the breadcrumbs. Place on a plate.
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Heat 2-3 inches of vegetables oil in a heavy saucepan, until it reaches 375 degrees. Once the oil is hot, carefully lower the croquettes into the oil, cooking 3-4 at a time. Let them cook for about 1-2 minutes, until golden brown, and then remove from the oil with a slotted spoon and place on a paper towel to absorb any excess oil.