►Warm 2 tablespoons olive oil in a large skillet over medium heat. Add onions, reduce heat to medium-low, and cook, stirring occasionally, until golden and caramelized, about 25 minutes. Season with salt and pepper. Stir in garlic, thyme, and red pepper flakes and cook for 1 minute.

►Add remaining 1 tablespoon olive oil and kale. Increase heat to medium-high and cook, tossing, until kale is wilted and tender, about 5 minutes. Transfer mixture to a bowl to cool.

►Melt butter in same skillet. Add flour, reduce heat to medium, and cook, whisking constantly, until a light golden paste forms. Whisk in 1 cup milk, season with salt and pepper, and cook, whisking, until mixture is very thick and no floury taste remains, 8 to 10 minutes. Remove from heat and whisk in heavy cream, sour cream, and remaining 1½ cups milk. Let cool for at least 10 minutes, then whisk eggs into mixture.

►Heat oven to 325°F. Generously grease a 9-inch-by-13-inch baking pan.

►Combine kale mixture, milk mixture, bread, butternut squash, and ½ cup smoked cheddar in a large bowl. Mix well and pour strata mixture into prepared baking dish. Let stand for 30 minutes, pressing down bread occasionally.

►Bake strata for 55 minutes to 1 hour, until almost set. Increase oven temperature to 475°F. Sprinkle remaining ½ cup cheese on top and bake for about 10 minutes more, until top is lightly browned. Let strata stand for 15 minutes before serving.