Savory Bread Pudding with Spinach, Chèvre, Smoked Ham

1 cup whole milk
4 cups 1-inch cubes crustless Italian or French bread
3/4 cup heavy cream
3 large eggs
2 garlic cloves, very finely chopped
1 teaspoon kosher salt
1/2 teaspoon ground black pepper
1/2 teaspoon freshly grated nutmeg
2 tablespoons chopped fresh sage leaves
1 tablespoon fresh thyme leaves
2 cups diced smoked ham (8 ounces)
1 (16-ounce) package frozen spinach, thawed and squeezed as dry as possible
6 ounces chèvre or other fresh goat cheese, coarsely crumbled (about 1 cup)
3 ounces Asiago cheese, coarsely grated (about 1 cup)
2 ounces Parmigiano-Reggiano or Pecorino Romano cheese, finely grated (about 1 cup)
Smoky Roasted Red Pepper Sauce, for serving (recipe follows)

Stir together the milk and bread in a large bowl. Let stand until the bread absorbs the milk, about 10 minutes. Meanwhile, whisk together the cream, eggs, garlic, salt, pepper, nutmeg, sage, and thyme in a medium bowl.

Spread half of the bread into a greased 2 1/2-quart shallow baking dish. Scatter half of the ham, spinach, chèvre, and Asiago over the bread. Drizzle half of the cream mixture evenly over the top. Add the rest of the bread to the dish. Scatter the remaining ham, spinach, chèvre and Asiago over the bread. Drizzle the rest of the cream mixture over it all. Sprinkle the Parmigiano-Reggiano over the top.

Cover the dish tightly with plastic wrap and refrigerate for at least 4 hours, preferably overnight.

Preheat the oven to 350°F. Uncover the dish. Bake until the pudding is firm in the center and browned around the edges, 50 to 60 minutes. Let sit for 10 minutes. Serve warm with a little of the sauce.

Smoky Roasted Red Pepper Sauce

4 tablespoons extra-virgin olive oil
2 cups chopped yellow onion
4 garlic cloves, chopped
2 cups drained and chopped roasted piquillo or other sweet red peppers
1 1/2 teaspoons smoked paprika, or to taste
1/4 cup sherry vinegar
1 tablespoon tomato paste
1 tablespoon packed brown sugar
Kosher salt and ground black pepper, to taste

Heat the oil in a large saucepan over medium-high heat. Add the onion and a pinch of salt and cook, stirring often, until very soft and golden, about 10 minutes. Add the garlic and peppers and cook, stirring often, for 5 minutes. Stir in the paprika, vinegar, tomato paste, and sugar.

Purée in a blender and return to the pan or purée directly in the pan with an immersion blender. Season with salt and pepper. Check the sauce for balance; it should be sweet, smoky, tangy, piquant, and bright.

Serve slightly warm or at room temperature. Store covered and refrigerated for up to one week. Stir well and check the seasoning before serving.