Preheat the oven to 450° F. Line a baking sheet with parchment paper or a silicone baking mat. Shred the cheese on a box grater and set aside.
In a large bowl, combine the flour, salt, and sugar. Add 1/2 cup of the cold butter to the flour mixture and use a pastry blender to cut the butter into pieces about the size of peas. Stir in the cheese and ham and mix well to combine. Pour about 3/4 cup of buttermilk into the bowl and mix well. If the dough is still too crumbly, gradually add more milk to get it to the point there you can roll it out, but not too sticky. Turn the dough out onto a floured surface. Sprinkle some additional flour over the dough. Knead the dough lightly being sure to fold it over on itself three to four times. Gently roll or pat the dough to about 3/4-inch thick. Use a 2 1/2" to 3" biscuit cutter to cut the biscuits by pressing strain down to cut them. Take care to not twist the cutter. Use a straight up and down motion. Transfer the biscuits to the prepared pan. Rework the dough and continue to cut biscuits until all the dough has been used.
Melt the remaining 2 tablespoons of butter and brush the tops of each biscuit. Bake 14 to 16 minutes, or until golden brown. Serve warm.
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