Heat the olive oil in a large Dutch oven over medium-high heat. Add the pork cubes and brown them on all sides, then remove the meat to a plate.
Lower the heat to medium and add the onion and carrots to the same pot. Cook until the onion turns soft.
Return the pork to the pot along with the apples, apple cider, chicken broth, and sage. Season with salt and black pepper. Bring the mixture to a simmer, then cover and cook for 45 minutes until the pork is tender.
While the stew simmers, stir together the flour, baking powder, and salt in a mixing bowl. Add the cheddar cheese, milk, and melted butter. Mix just until a soft dough forms.
Drop spoonfuls of the dough onto the surface of the simmering stew. Cover the pot and cook for 15 minutes without lifting the lid so the dumplings steam through.