Add butter to a medium saucepan and set over medium heat. Once butter has melted, add shallots and garlic and cook until fragrant, about 1 minute. Add 1/2 cup of flour and stir with a wooden spoon until a ball forms.
Slowly add milk to flour mixture and stir constantly. Once smooth, remove from heat and add prosciutto ham and Manchego cheese and combine. Add a couple pinches of black pepper. Mixture should be thick, like the consistency of dough. Transfer mixture to a storage container and refridgerate until chilled, at least 2 hours. I did mine overnight just to make sure it was chilled through.
Once dough is chilled, remove from refrigerator and place onto floured work surface. Roll dough into a long log, about 24 inches long and 1 inch in diameter. Cut dough into 12 equal pieces (if you want smaller bites, cut into 24 pieces).
Arrange coating ingredients into three separate bowls: one for flour, one with 2 scrambled eggs, and one with panko bread crumbs. Dip one of the dough pieces into flour and shake off excess. Coat in egg mixture and then place in bread crumbs, pressing crumbs into the dough to make sure they're secure. Form dough into a ball or a log. Place on a baking sheet. Repeat with other dough pieces.
Place prepare croquettes in the refrigerator to get solid again, about 30 minutes. When you have about 15 minutes left, begin to preheat your frying oil. Add frying oil to a cast iron pan (oil should not be any higher than halfway up the pan). Place over medium heat.
Remove croquettes from the fridge. Preheat oven to 350 degrees. Test one croquette in the hot oil. If it begins to sizzle, the oil is hot and ready to use. If it does not sizzle, give it a few more minutes.
Once the oil is ready, cook 4 croquettes in the oil for 2 minutes (if you did smaller croquettes, you will cook for only a minute or so). Flip and cook for another 1-2 minutes (1 minute if croquettes are small), until golden brown. Remove from oil and dry on paper towels. Repeat with the rest of the croquettes, making sure not to overcrowd the pan (4 large croquettes or 8 small croquettes at one time, maximum).
Once all the croquettes are fried, finish them off in the oven so they are cooked through, about 2-3 minutes. Serve immediately.