For the strata: Preheat the oven to 325 degrees F.
Put the bread on a baking sheet and toast until lightly golden brown on both sides, about 10 minutes. Remove and let cool.
Combine the butter and oil in a large saute pan over medium heat and cook until it begins to shimmer. Add the onions, toss to coat in the butter, and then add the sugar and season with salt and pepper. Cook the onions, stirring occasionally, until golden brown and caramelized, about 30 minutes. Remove from the heat and let cool slightly.
In a large bowl, combine the bread cubes, onions and spinach. In another large bowl, whisk together the eggs, 1 1/2 cups cheese, milk, heavy cream, mustard, 1 teaspoons salt, 1/2 teaspoon pepper, thyme and nutmeg.
Butter a 9 by 13-inch decorative baking dish. Add the bread cubes to the baking dish, and then pour the egg mixture over the bread cubes and press down to make sure the bread is totally submerged.
Preheat the oven to 350 degrees F.
Bake the strata uncovered until the mixture has puffed up slightly, is golden brown on top and doesn't shimmy with uncooked custard when you shake the pan, about 1 hour. Sprinkle the remaining 1/2 cup cheese over the top during the last 10 minutes of baking. Let cool 5 minutes before serving.
For the tomatoes: Heat oil in a large saute pan over high heat until it begins to shimmer. Add the tomatoes, season with salt and pepper and cook until the tomatoes softened. Add the chives and remove from the heat.
Serve large spoonfuls of the strata onto plates and top with sauteed tomatoes.