In a 6-quart Dutch oven melt butter over medium heat. Add onion, celery, and carrot; cook 7 minutes or until vegetables are soft, stirring occasionally. Add broth and sweet potatoes. Bring to boiling; reduce heat. Simmer, uncovered, 8 to 10 minutes or until sweet potatoes are tender.
In a bowl whisk together half-and-half and flour; add to soup mixture along with ham. Cook and stir until slightly thickened and bubbly. Cook and stir 1 minute more. Stir in cheese until melted and smooth. Top with green onions and serve with Buttermilk Biscuits.
Preheat oven to 450 degrees F. In a large bowl combine 3 cups all-purpose flour, 1 tablespoon baking powder, 1 tablespoon sugar, 1 teaspoon salt, and 3/4 teaspoon cream of tartar. Using a pastry blender, cut in 3/4 cup butter (or 1/2 cup butter plus 1/4 cup shortening) until mixture resembles coarse crumbs. Make a well in center of flour mixture. Add 1 1/4 cups buttermilk all at once. Using a fork, stir just until moistened. Turn dough out onto a lightly floured surface. Knead dough by folding and gently pressing it just until dough holds together. Pat or lightly roll dough until 3/4-inch thick. Cut dough with a floured 2 1/2-inch biscuit cutter, rerolling scraps as needed and dipping cutter into flour between cuts. Place dough circles 1 inch apart on ungreased baking sheet. Bake 10 to 14 minutes or until golden. Serve warm. (Wrap leftover biscuits in foil and store up to 24 hours. Reheat wrapped biscuits in a 300 degrees F oven before serving.) Makes 12.