Put the bacon in a large cold skillet then cook over medium heat, stirring occasionally, until crispy, about 10 minutes. Transfer to a paper towel-lined plate with a slotted spoon. Discard all but about 2 tablespoons of the bacon drippings. Increase the heat to medium high, add the kale and cook, stirring occasionally, until tender, about 5 minutes. Set aside.
Whisk the half-and-half, Parmesan and eggs together in a bowl with 1 1/2 teaspoons salt and a few grinds of pepper.
Layer half of the bread cubes in the prepared baking dish. Sprinkle with 1 1/2 cups of the cheese, followed by the bacon, kale and sundried tomatoes. Cover with the remaining bread cubes and pour the custard evenly over top. Cover with plastic wrap and refrigerate overnight. (This can also be assembled the day of; just let soak at room temperature 30 minutes to 1 hour before baking.)
On the day of baking, preheat the oven to 325 degrees F. Let the strata sit at room temperature while the oven preheats. Remove the plastic and sprinkle with the remaining 1/2 cup of cheese. Bake until slightly puffed and just no longer jiggly in the center, 40 to 50 minutes. Let rest 5 minutes before serving.
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