Heat oil in a large saucepan over medium-high heat. Add carrots and onion. Cook, stirring occasionally, 3 to 4 minutes. Add stock and 2 cups water; bring to a simmer. Cover, and cook until vegetables are just tender, 8 to 10 minutes.
Meanwhile, combine flour, cornmeal, baking powder, and salt in a medium bowl. Stir in cheese and corn. Add egg and milk; stir vigorously to make a slightly sticky dough. Using lightly floured hands, form dough into 1-inch balls; set aside.
Add the turkey to soup; season with salt and pepper. Bring to a gentle boil; add dumplings. Cover; cook until dumplings are cooked through, 5 to 6 minutes. Serve warm.