Preheat oven to 375°F (190°C).
Kale Pesto: In a blender or with a hand immersion blender, purée kale, basil, oil, lemon juice, nuts and garlic. Stir in cheese.
Strata: Scrape pesto into a large bowl; toss with cubed bread to coat. Arrange half the pesto and bread mixture in a large 10 to 12-inch (25 to 30 cm) greased cast iron skillet. Top with half the chicken, cheese and red pepper. Repeat steps to make the second layer.
In a large bowl, whisk eggs; add milk and whisk to combine. Pour over filling in skillet. Bake for 35 to 40 minutes until puffed and golden. Let cool 10 minutes before serving. Season with pepper to taste.
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