Preheat the oven to 400F. Line a large baking sheet with parchment paper. Place the stick of butter in the freezer for 10 minutes.
In a large bowl, combine all-purpose flour, brown sugar, salt, black pepper, baking powder, and baking soda.
In a small bowl, whisk the egg. Add in Greek yogurt and milk, whisk to combine.
Pour the wet ingredients into the large bowl with the dry ingredients, and mix to combine with a rubber spatula.
Now, remove the butter from the freezer and grate it into the dough, and knead it to spread it around.
Finally, fold in cheddar cheese, ham, and green onion gently into the shaggy dough. Use a rubber spatula.
Place your shaggy dough on a floured surface. Cover both sides of the dough with flour and form an 8-inch circle with your hands. Cut the dough into 8 triangles and carefully transfer them onto the baking sheet.
Bake for 18-20 minutes until the scones turn light golden.
Let them cool down for 5 minutes on the baking sheet and serve warm.