
In a large skillet, heat 1 tablespoon olive oil over medium-high heat. Add the ham, onion, and bell pepper, and cook, stirring, until softened, 8 to 10 minutes. Add the salt, pepper, and thyme, and cook for 1 minute more. Remove from the heat and set aside to cool slightly.
Brush a 13-by-9-inch baking dish with the remaining 1 tablespoon olive oil. Add the croissant pieces, top with the ham mixture, and mix gently to combine.
In a medium bowl, whisk together the milk, half-and-half, hot sauce, and eggs. Add half of both cheeses. Pour the mixture over the croissants and mix gently to combine and evenly distribute. Cover and refrigerate for at least 1 hour and up to 4 hours.
Preheat the oven to 350°F. Remove the casserole from the refrigerator, uncover, and let sit at room temperature for 15 minutes before baking, pressing down on the croissants with a spatula to make sure they are well soaked.
Sprinkle the casserole with the remaining cheeses. Bake until puffed, golden brown, and the cheese is melted, about 45 minutes. Let rest 15 minutes before serving.