Preheat the oven to 375 degrees F.
Make the Biscuits: Using a sieve over a bowl, sift together the flour, baking powder, sugar, and salt. Repeat. Add the butter and shortening and, using your fingertips, mix until crumbly. Add the buttermilk and mix with a wooden spoon until the dough just comes together. (Don't over mix!)
Transfer the dough to a lightly floured work surface and, using a rolling pin, roll out to 1/2 to 3/4-inch thick rectangle. Using a 3 to 4-inch round cutter, punch out biscuits and transfer them to a parchment paper-lined baking sheet. Gently press the dough scraps into another rectangle and punch out more biscuits. Discard any scraps. Lightly dust the tops of the biscuits with flour and bake until golden brown, about 12 to 15 minutes. Let the biscuits cool slightly. (You'll only need 6 biscuits for the sandwiches, so serve the rest with soup, spread with butter and jam, or just freeze for another day.)
Make the Sandwiches: Preheat the broiler. Split 6 of the biscuits in half and spread the top halves with 1 tablespoon of tomato jam each.
Divide the ham among the bottom halves of the biscuits and sprinkle the cheese over the top.
Arrange the bottom halves a baking sheet and place under a broiler until the cheese melts. Cover with the biscuit tops and serve immediately.
1/4 cup extra-virgin olive oil
1/2 Spanish onion, finely chopped
4 cloves garlic, finely chopped
Leaves from 2 rosemary sprigs, finely chopped
Pinch red pepper flakes
4 cups peeled, seeded, and roughly chopped plum tomatoes
Kosher salt and freshly ground black pepper
Heat 2 tablespoons of oil in a heavy bottomed saucepan over medium heat. Add the onion, garlic, rosemary and pepper flakes and cook, stirring, until soft and fragrant. Add the tomatoes and cook, stirring occasionally, until the tomatoes have given up most their water and turned a deep orange red, about 15 to 20 minutes. Whisk in the remaining 2 tablespoons of oil and season with salt and pepper, to taste.