Butter a 3-quart gratin dish or casserole. Whisk eggs, milk, cream, 1 teaspoon salt, 1/4 teaspoon black pepper, nutmeg, and cayenne pepper in a large bowl. Toss ham, tomatoes, parsley, and chives together in a small bowl.
Scatter 1/2 cup of cheese into prepared dish. Layer with 1/3 of bread and half of ham mixture. Repeat. Top strata with a final layer of bread and remaining 1 cup cheese.
Pour egg mixture over top of dish and gently press to make moistened all bread layers. Drizzle with melted butter, cover strata with plastic wrap, and refrigerate for 1 hour or up to overnight.
Preheat to 350 degrees F. Bake strata, on middle oven rack, uncovered, until slightly puffed and golden brown, about 45 minutes. Let rest 10 minutes before serving.
Preheat oven to 250 degrees F.
Arrange tomatoes, cut side up, in a single layer on a parchment-lined baking sheet. Sprinkle with salt. Bake until slightly shriveled but still plump, about 5 hours.
If not using immediately, refrigerate dried tomatoes, covered, up to 4 days. Or, if desired, layer tomatoes with herb sprigs and add olive oil. Refrigerate, covered, for up to 2 weeks.
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