Preheat the oven to 425°F. Line a baking sheet with parchment paper or a silicon mat. Dice enough ham into uniform pieces to make 1 cup total. Set aside.
Whisk the flour, salt, baking powder, and sugar together. Using your fingers or a pastry blender, work the butter into the flour until the mixture is crumbly.
Mix in the cheese, green onions, and ham until evenly distributed.
Add 3/4 cup of the cream, mixing with your fingers to combine. Press and squeeze the dough together, add some more cream until the dough comes together. Leave at least 2 tablespoons to brush on before baking. On a well-floured work surface pat the dough into a smooth 7" disk about 3/4" thick. Move the dough disk to the baking sheet. Use a knife or pizza cutter to cut the disk into 8 wedges, spreading the wedges apart a bit on the baking sheet.
Brush the scones with a bit of cream. Bake for 20-22 minutes, until they're golden brown. Remove them from the oven, and cool. Serve warm, or at room temperature with or without butter (optional).