Butter a 2 1/2- to 3-quart baking dish. In a large bowl, whisk together the eggs, milk, mustard, paprika, and salt.
Place half of the bread in the bottom of a baking dish. Top with half each of the ham, scallions, and cheese. Repeat with a second layer of the remaining ham, scallions, and cheese, pressing down as necessary to fit in the baking pan.
Carefully pour the egg mixture over into the bread. Cover and refrigerate overnight.
Preheat the oven to 350 degrees Fahrenheit. Pull strata out of fridge and let come to room temperature for 15 minutes. Bake the strata, uncovered, until puffed and lightly golden brown, about 1 hour to 1 hour and 20 minutes. Let rest for 5 minutes before serving.