Prepare the ingredients by dicing the ham and shredding the cheese.
In a mixing bowl, combine the diced ham, shredded cheese, flour, garlic powder, salt, and black pepper.
In a separate bowl, whisk the eggs until well beaten and smooth.
Form the mixture into sticks, about 2 inches long and 1 inch thick.
Heat vegetable oil in a frying pan over medium heat until hot.
Fry the formed sticks in batches, turning occasionally, until golden brown.
Remove the sticks from the oil and drain them on paper towels to remove excess oil.
Serve the ham and cheese sticks warm with your selected dipping sauce.