In a medium pot, cover the potatoes with cold water by an inch or two, add the tablespoon of salt, and bring to a boil. Turn down the heat to maintain a brisk simmer and cook until potatoes are tender, about 15 minutes give or take depending on how big the pieces are. Drain potatoes in a colander and set aside.

While the potatoes cook, prepare the shallots and cabbage.

Return the same pot to the stove and heat the butter and olive oil over medium-high heat. Add the shallots and cabbage and cook, stirring frequently (tongs are useful here), until cabbage is wilted and somewhat tender, about 5 minutes.

Add the milk, raise heat to high, and warm until gently simmering. Remove pot from heat. Return the potatoes to the pot, add the teaspoon of salt and the pepper, and mash well with a potato masher. Stir in the cheddar.

At this point the mixture will need to cool slightly — so either turn it out into a mixing bowl to speed the process or leave it in the pot to spare a dirty dish and wait a few minutes longer.

Make a well in the center of the potatoes and crack the eggs into it. Beat them well, then stir completely into the mix. Sprinkle the flour over the whole thing and mix in thoroughly.

Heat a thin layer of oil over medium-high heat in a 12-inch nonstick skillet. When it shimmers, spoon about three tablespoons of the mixture into the pan to form a reasonably round cake. You should be able to fit 4 cakes in the skillet at a time without crowding. Cook undisturbed for a few minutes, until the underside is nicely browned. Flip gently with a spatula and cook until the second side is browned, a few minutes more. You may need to adjust the heat so the cakes cook through in the same amount of time they take to brown.

Set on a paper towel-lined plate or in a 200°F oven while you cook the remaining batches. Serve warm or at room temperature.