Whisk together the eggs very well in a large bowl with milk or half-and-half. Whisk in mustard and thyme and season with salt and pepper. Note that the cheese and ham bring a lot of salt to the mix. I often make this strata without salt, so at the very least use a light hand. But absolutely do add black pepper, which adds balance to the dish. Fold in cubes bread and allow to sit while you preheat oven.

Position rack in middle of oven. Preheat oven to 350°F/180°C. Coat the inside of a 13 x 9-inch (33 cm x 23 cm) casserole dish with nonstick spray.

Fold ham, cheeses and tomatoes into strata mixture and scrape into prepared pan.

Bake for about 35 to 45 minutes or until custard is set. The strata will puff up a bit and exposed pieces of bread will get crispy here and there, which is perfect. Cool pan on a rack for 5 minutes and serve warm or at room temperature. The strata can be refrigerated but should be warmed at least up to room temperature, which you can do in the microwave or a low oven.