If you are making the strata right away and not letting it sit overnight, preheat the oven to 400 degrees F. Grease the bottom of a 9x13 baking dish with cooking spray or butter.
Heat olive oil in a large skillet over medium heat. Finely chop the shallot, and prepare the kale leaves by taking them off their stems and roughly chopping them. Add the shallot and kale to the skillet and cook for 7 to 10 minutes, or until the shallots are tender and the kale leaves are wilted.
Transfer the cooked kale and shallots to the baking dish, and then add the torn or cubed sourdough bread, followed by the shredded gruyere and cheddar cheeses.
In a large bowl, whisk together the eggs, milk, mustard, salt, and pepper. Pour this mixture over the sourdough, veggies, and cheese. Gently shake the dish with your hands to promote an even distribution of the egg mixture. You may also want to use a spoon to move the kale around. Sprinkle a little more cheese on top if desired. Cover with foil.
Put the covered strata in the fridge for at least 20 minutes so the bread can soak up everything. You can keep it in the fridge overnight if you plan to bake the strata the following morning.
Bake the strata for 30 to 35 minutes covered in foil. Remove the foil and bake it for another 15 to 20 minutes until you see the eggs set in the middle and bubbling on the edges. Enjoy warm!