Coat a 1 1/2-quart baking dish with butter.
In a large bowl, whisk the eggs, milk, salt, paprika, and pepper until combined. Fold in the greens, mushrooms, tomatoes, red onion, ham, bread, and cheddar and Havarti cheeses.
Pour the mixture into the baking dish and top with the Parmigiano-Reggiano cheese. Cover the baking dish with plastic wrap and set in the refrigerator to rest for 1 hour or overnight.
When ready to bake, preheat the oven to 350°F (175°C) with a rack placed in the center of the oven. Remove the strata from the refrigerator to rest at room temperature while the oven preheats.
Remove the plastic wrap, cover the strata with foil, and bake for 35 minutes. Uncover and bake until it is puffed and set in the center, with an internal temperature of at least 165°F (74°C), 15 to 25 minutes more. Let rest for 5 minutes before slicing.
Garnish with Parmigiano-Reggiano and the parsley, if desired. Enjoy warm.