Preheat the oven to 325°F. Spray an 8x8-inch baking dish with non-stick cooking spray.
Onto a baking sheet, spread the bread cubes in an even layer. Bake for 10 minutes, until lightly crisp. Set aside.
Into a large skillet over medium-high heat, crumble the sausage and cook through, using a spatula to break up any large clumps. Set aside.
In a large bowl, stir together the toasted bread cubes, cooked sausage, kale, and cheddar cheese. Transfer to the prepared baking dish.
In a medium bowl, combine liquid vegan egg, milk, Dijon mustard, rosemary, salt, and pepper. Pour over the bread cube mixture in the pan
Cover the pan and set in the refrigerator for 8 hours or up to overnight.
Remove from the refrigerator and preheat the oven to 375°F.
Lightly cover with foil and bake for 45 minutes, uncovering the strata during the last 20 minutes.
Allow the strata to cool to a warm room temperature then slice and serve.