Preheat your oven to 375°F. Line a baking sheet with parchment paper.
In a mixing bowl, combine the flour, baking powder, salt, and black pepper. Mix well.
In another bowl, cream together the softened butter and shredded cheddar cheese until smooth. Add the diced ham and mix until evenly combined.
In a separate small bowl, whisk together the milk and egg. Gradually add this mixture to the cheese and ham mixture, stirring until just combined.
Slowly add the dry ingredients to the wet mixture, stirring until a dough forms. Do not overmix.
Turn the dough out onto a floured surface and roll it out to about 1/2 inch thick. Cut into strips or desired shapes.
Place the cut dough onto the prepared baking sheet, spacing them about 1 inch apart. Bake in the preheated oven for 15-20 minutes, or until golden brown.
Allow to cool slightly before serving.