Heat the oven to 425°F with a rack in the upper middle position.

Set a 12-inch oven-safe skillet or a medium Dutch oven on the stove. If you do not have an oven-safe pan, lightly oil a 2-quart baking dish and keep it nearby.

Add the olive oil and butter to the skillet over medium heat.

When the butter melts, add the sliced onions and 1/2 teaspoon of the salt to the skillet. Stir well so the onions are coated with the fat.

Cook the onions for 12 to 15 minutes, stirring every few minutes, until they are very soft and deep golden brown. If they start to darken too fast, lower the heat a little.

Add the sliced mushrooms and the black pepper to the skillet. Stir to mix them with the onions.

Cook the mushrooms for 6 to 8 minutes, stirring now and then, until they give off their liquid and the pan looks mostly dry again.

Add the thyme leaves and Dijon mustard to the skillet. Stir for about 30 seconds, just until the mustard is worked through the onions and mushrooms.

Add the drained cannellini beans and the vegetable broth to the skillet. Stir gently so the beans stay mostly whole.

Bring the mixture to a gentle simmer over medium heat. Let it simmer for 5 minutes, until the liquid reduces slightly but still bubbles around the beans.

Taste the bean mixture and add the remaining 1/2 teaspoon salt if needed. Remove the skillet from the heat.

If you are using a baking dish instead of an oven-safe skillet, transfer the hot bean mixture into the prepared baking dish now.