Savory smoked ham, spinach, and cheese bread pudding

8 cups lightly packed small, tender spinach (about 1 pound);

1 cup whole milk;

4 cups 1-inch cubes crustless day-old Italian or French bread;

3⁄4 cup cream;

3 large eggs;

2 garlic cloves, finely chopped;

1 teaspoon kosher salt;

1⁄2 teaspoon pepper;

1⁄2 teaspoon freshly ground nutmeg;

2 tablespoons chopped fresh sage;

1 tablespoon chopped fresh thyme;

2 cups diced smoked ham (about 8 ounces);

3⁄4 cup soft, fresh goat cheese (chèvre);

1⁄2 cup grated Asiago cheese, divided;

3⁄4 cup grated Parmesan cheese.

Fill a large bowl with ice water. Bring a large saucepan of water to a boil. Add 1⁄2 teaspoon kosher salt per cup of water. Working in batches, place the spinach in a large strainer and lower it into the boiling water until it wilts, about 3 seconds. Quickly transfer the spinach into the ice water to stop the cooking and set the bright green color. Drain the spinach and squeeze it as dry as possible. Coarsely chop the spinach.
Combine the milk and bread cubes in a large bowl and let stand until the bread absorbs the milk, about 10 minutes.
Whisk together the cream, eggs, garlic, salt, pepper, nutmeg, sage, and thyme in a medium bowl.
Spread half of the bread mixture in a lightly greased 21⁄2-quart baking dish. Layer on the ham, the spinach, the goat cheese, and 1⁄4 cup of the Asiago. Top with the remaining bread mixture. Slowly pour the cream mixture evenly over the top and sprinkle with the remaining 1⁄4 cup Asiago and the Parmesan. Cover the dish with plastic wrap and refrigerate for at least 4 hours and preferably overnight.
Preheat the oven to 350°F. Bake the pudding uncovered until it is firm in the center and brown around the edges, 50 to 60 minutes. Let sit for 10 minutes before serving.