In a medium pot of salted water bring potatoes to a boil. Reduce heat and simmer until tender, about 10 minutes. Drain thoroughly and mash until smooth (or process through a potato ricer). Stir in milk, butter, and salt, and let cool slightly. Stir in Manchego, ham, and 1 egg, lightly beaten. Roll mixture into 20 walnut-size balls and place on a baking sheet. Refrigerate until firm, about 1 hour.

Place flour and bread crumbs on two separate plates. In a shallow bowl, whisk remaining 2 eggs with 1 tablespoon water and set bowl beside flour and bread crumbs. Roll croquettes in flour, dip in egg wash, and roll in bread crumbs.

In a medium skillet over medium-high heat, heat 1 inch of oil (about 2 cups). Working in batches, fry croquettes until golden and crisp, about 2 minutes per side. Transfer to a paper towel–lined plate to drain. Sprinkle with flaky sea salt. Repeat with remaining croquettes, adding additional oil as needed.