Prepare the pan. Grease a metal 8-inch baking pan with the olive oil. Lay the bread cubes, kale, Cheddar and sausage slices in the pan and toss them together.
Combine the wet ingredients. In a large bowl, whisk together the milk, mustard, rosemary, eggs, 1 teaspoon salt and 1/2 teaspoon pepper and until thoroughly combined. Drizzle over the contents of the pan. Cover the pan and put it in the refrigerator to rest 8 hours or up to overnight.
Preheat the oven to 375 degrees F. Pull the strata out of the refrigerator and uncover it.
Bake the strata. Bake the strata until the egg custard is set and the edges are browned, about 45 minutes. Allow the strata cool to a warm room temperature, about 15 minutes, before serving.
To make bread cubes, cut slices of bread into three pieces lengthwise, then 3 slices down. If you want to toast them first, to give the bread in the strata a little bite, toss them on a baking sheet and put them in a 400 degrees F oven for about 10 minutes.