Preheat oven to 375 degrees.

In a medium skillet, melt the butter over medium heat. Add the onions, and season lightly with salt and pepper. Lightly stir onions to evenly coat in butter. Cook for about 4 minutes, until onions are soft, then lower the heat to medium-low. Stir onions frequently until they are browned, about 35 minutes. Transfer to a small bowl, and return the skillet to the stove top.

In the skillet, heat the olive oil over medium-high heat. Add the mushrooms and allow to brown, 10-12 minutes, stirring occasionally until they become tender.

Mix in the thyme leaves and kale—cook for about 4 minutes more (until kale wilts slightly), then add in the white wine. Stir until almost completely evaporated, 2-4 minutes. Remove from the heat and return the caramelized onions to the pan. Allow to cool for 5-7 minutes.

In a large bowl, whisk the eggs, egg yolk, milk, nutmeg, and cayenne; season with salt and pepper. Add the bread cubes and gently push into milk mixture. Next, add mushrooms and 1 1/4 cup of the cheese—gently mix with hands to combine.

Transfer the bread pudding to the prepared baking dish and top with the remaining cheese.

Cover pan with foil and bake on the center rack for about 25 minutes. After time is up, remove foil from pan and bake for another 10 minutes.

Serve hot!