Heat the oil to 365 to 375 degrees F in a cast-iron Dutch oven over medium heat.
In a large bowl, mix together the flour, cheese, scallions, sugar, baking powder, salt and baking soda. In a smaller bowl, whisk together the buttermilk, sour cream and egg. Pour the wet ingredients into the dry and mix to combine, making sure there are no dry bits left.
Turn the dough out onto a floured surface and lightly coat with flour. Pat the dough into a 1-inch disc. Then, using a large glass dipped into flour, cut out rounds for the doughnuts. Press the dough back together to cut out another doughnut or 2. Dip your finger into the flour, then press it into the center of each doughnut while it is on the floured surface to create a hole. Move your finger around in the hole to make it larger. Turn the doughnut over and repeat on the other side to make sure the hole is clear and large.
When the oil is ready, place the doughnuts into the oil carefully and fry for 3 to 4 minutes per side. The doughnuts are thick and take time to cook all the way through. You can flip them every minute to keep them from browning too much on each side. Transfer the doughnuts to a paper towel-lined baking sheet and sprinkle with a little salt. Serve immediately.