Preheat your oven to 350°F (175°C).
Heat 1 tablespoon of olive oil in a 10-inch cast iron skillet over medium heat. Add the bread cubes and season with salt and pepper. Toast, stirring frequently, until golden and crisp in spots, about 3–4 minutes. Transfer to a plate.
In the same skillet, heat the remaining olive oil over medium-low. Add the sliced leeks, minced garlic, and thyme. Cook for 3–4 minutes until softened and fragrant. Remove from heat, then stir in the spinach and toasted bread cubes.
In a large mixing bowl, whisk together the eggs and milk until smooth. Stir in the cottage cheese and half the Parmesan cheese. Season with a pinch of salt and pepper.
Pour the egg mixture over the vegetables and bread in the skillet, gently tossing to coat evenly. Sprinkle the remaining Parmesan on top.
Slide the skillet into the oven. Bake for 30–35 minutes, or until the strata is puffed, golden, and the eggs are fully set. Let cool for at least 5 minutes before serving.