Set a rack in the middle of the oven and heat the oven to 425°F.

Place a deep 12-inch ovenproof skillet on the counter so it is ready for both the stovetop and the oven.

Prepare the leeks if you have not already: trim off the root end and the dark green tops, split them lengthwise, rinse away any grit between the layers, and thinly slice them.

Drain the butter beans in a colander, rinse them under cold water, and let them sit to drain while you start the skillet.

Slice the potatoes into very thin rounds, about 1/8 inch thick, so they soften quickly in the sauce.

Put the olive oil and butter into the skillet and warm them over medium heat until the butter melts.

Add the sliced leeks and 1/2 teaspoon of the salt to the skillet. Cook for 5 to 7 minutes, stirring now and then, until the leeks are soft and glossy but not browned.

Add the Dijon mustard and half of the thyme leaves to the skillet. Stir for about 30 seconds so the mustard loosens and coats the leeks.

Pour the half-and-half and 1/2 cup water into the skillet. Add the remaining 1 teaspoon salt and the black pepper, and bring the liquid to a gentle simmer over medium heat.

Add the sliced potatoes to the skillet, spreading them out so most of the slices are touched by the liquid.

Keep the skillet at a gentle simmer for 10 to 12 minutes. Stir once or twice carefully so the potatoes do not stick, and cook until the slices are starting to bend and a knife meets only a little resistance.

Add the drained butter beans to the skillet and fold them into the potatoes and leeks gently so the beans stay mostly whole.