Gather all your ingredients: flour, baking powder, salt, black pepper, cheddar cheese, diced ham, milk, egg, and melted butter.
In a mixing bowl, combine the all-purpose flour, baking powder, salt, and black pepper. Stir until well mixed. Add the shredded cheddar cheese and diced ham, mixing evenly.
In a separate bowl, whisk together the milk, egg, and melted butter. Pour the wet ingredients into the dry mixture and stir gently until just combined.
Turn the dough onto a lightly floured surface and knead gently for about 1. Knead until everything is combined without making it tough. Roll the dough into a rectangle about 1/2 inch thick.
Cut the dough into strips approximately 1 inch wide and 4 inches long. Place them on a baking sheet lined with parchment paper, spaced about 1 inch apart.
Preheat your oven to 400°F. Bake the ham and cheese sticks for 15 to 18 minutes, or until golden brown.
Remove the sticks from the oven and let them cool slightly before serving.