Preheat the oven to 350 degrees F. Toss the tomatoes with the thyme, olive oil and 1/2 teaspoon each salt and black pepper in a bowl. Spread on a baking sheet and roast, stirring once or twice, until the tomatoes are slightly dried and browned in spots, 1 hour to 1 hour, 15 minutes. Cool completely.
Whisk the milk, eggs, fontina, 3/4 cup parmesan, the cayenne, 1 teaspoon salt and 1/4 teaspoon black pepper in a large bowl. Add the bread cubes and greens and toss to coat. Pour into a 3-quart baking dish, then top with the roasted tomatoes. Cover and refrigerate overnight.
The next day, preheat the oven to 350 degrees F. Sprinkle the strata with the remaining 1/4 cup parmesan and bake, uncovered, until golden brown and almost set, about 45 minutes. Turn off the oven, but keep the strata inside 10 minutes before serving.