1. Preheat the oven to 325°F. Set a shallow 8-inch square or 11 × 7-inch baking dish inside 13 × 9-inch baking pan. Set a kettle of water on to heat.

2. Heat a large skillet over medium heat; add the oil. When it is hot enough to sizzle a mushroom, add all of the mushrooms; cook, stirring, for 5 minutes. Add the onions cook over medium-low heat until the onions are tender and the mushrooms are golden brown, about 5 minutes.

3. Meanwhile, finely chop the parsley, thyme, rosemary, and garlic together. Add to the mushrooms: cook for 1 minute. Stir in the roasted red peppers, 1 teaspoon salt, and a generous grinding of black pepper. Remove from the heat.

4. Add the rice, breaking up any clumps with the side of a spoon; stir to blend. Let cool slightly. Gently stir in half of the goat cheese; transfer to the baking dish. Sprinkle the remaining goat cheese on top.

5. Whisk the eggs and yolks in a large bowl until blended. Gradually whisk in the milk. Add 1 teaspoon salt and a grinding of black pepper. Carefully pour over the mushroom and rice mixture.

6. Place on the center oven rack. Carefully add enough very hot water to the baking pan to come halfway up the sides of the baking dish. Bake until the custard is almost set in the center and the tip of a knife inserted just off center comes out clean, about 1 hour and 15 minutes. Let the pudding stand in the hot-water bath for 15 minutes before serving. (The center will continue cooking and setting as the pudding stands.)