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.
Meanwhile, heat the butter and canola oil in a large saute pan over medium heat until shimmering. Add the onions and toss to coat, then add the sugar and season with salt and pepper. Cook, stirring occasionally, until the onions are golden brown and caramelized, about 20 minutes. Remove from the heat and let cool slightly.
In a large bowl, combine the toasted bread, caramelized onions and spinach. In another large bowl, whisk the eggs, milk, heavy cream, 1 1/2 cups cheese, the mustard, 1 teaspoon salt, 1/2 teaspoon pepper, the thyme and nutmeg. Butter a 9-by-13-inch baking dish. Add the bread mixture to the baking dish, then pour the egg mixture over the bread and press down to make sure it is totally submerged.
Increase the oven temperature to 350 degrees F. Bake the strata, uncovered, until it is golden brown on top, puffed slightly and doesn't shimmy with uncooked custard when you shake the pan, 50 minutes to 1 hour; sprinkle with the remaining 1/2 cup cheese during the last 10 minutes of baking. Let cool 5 minutes before serving.
Meanwhile, prepare the tomatoes: Heat the canola oil in a large saute pan over high heat until it begins to shimmer. Add the cherry tomatoes, season with salt and pepper and cook until softened, 4 to 5 minutes. Add the chives and remove from the heat. Serve the strata with the sauteed tomatoes.