Preheat the oven to 375°F. Grease a baking sheet or line it with parchment paper.
In a mixing bowl, combine the flour, baking powder, salt, and black pepper.
Stir in the shredded cheddar cheese and diced ham until evenly distributed.
In a separate bowl, whisk together the milk, egg, and melted butter.
Pour the wet ingredients into the dry ingredients and mix until just combined. Do not overmix.
Scoop the dough onto the prepared baking sheet, forming small sticks about 3 inches long and 1 inch wide, spacing them about 2 inches apart.
Bake for 15-20 minutes, or until golden brown.
Remove from the oven and let cool slightly before serving.