Whisk together the eggs, half-and-half, 1 teaspoon salt and 1/4 teaspoon pepper in a large bowl. Toss together the ham, sun-dried tomatoes and chives in a small bowl.
Sprinkle 1/2 cup of the Gruyere into the bottom of the prepared dish. Layer with 1/3 of the bread cubes and half of the ham mixture. Repeat the Gruyere, bread and ham layers, ending with a third layer of bread cubes and then topping with the remaining 1 cup Gruyere.
Pour the egg mixture over the top and gently press with a spatula to moisten the bread layers. Drizzle with the melted butter, cover with plastic wrap and refrigerate for at least 1 hour or up to overnight.
Preheat the oven to 350 degrees F.
Remove the plastic wrap from the dish and bake the strata, uncovered, until slightly puffed and golden brown, about 45 minutes. Let stand at least 10 minutes before slicing.
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