Put the mushrooms on a baking sheet, drizzle with olive oil and sprinkle with salt and pepper. Roast until golden brown, 15 to 20 minutes. Set aside.
Mix together the milk, half-and-half, oregano, eggs and some salt and pepper in a large pitcher or bowl. Set aside.
Set a large skillet over medium heat and add the sausage patties. Cook until golden brown, 4 to 5 minutes per side. Set aside to cool, then chop into cubes.
In the same skillet from the sausage, heat some olive oil over medium-high heat. Throw in the kale and cook until slightly wilted, 2 minutes.
Grease a lasagna pan with some butter. Layer half the bread, half the kale, half the mushrooms, half the sausage and half the cheese in the prepared pan. Repeat with the other half of the same ingredients, ending with the cheese. Slowly pour the egg mixture all over the top. Cover with plastic wrap and refrigerate overnight.
Twenty to thirty minutes before baking, remove the lasagna pan from the fridge. Preheat the oven to 350 degrees F.
Replace the plastic wrap on top of the strata with aluminum foil. Bake for 30 to 40 minutes, then remove the foil and continue to bake until the top is golden brown and slightly crisp, 10 to 15 minutes more (keep a close eye on it).