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1
Heat the oven to 400u00b0F. Dump the brussels sprouts into a bowl, sprinkle with salt and pepper and the red pepper flakes, and drizzle with enough olive oil and mustard to coat. Give the brussels sprouts a good toss with your hands, making sure they're well-coated with oil and mustard. Spread evenly onto a parchment-lined baking sheet. Roast, stirring once for even browning, until crispy and cooked through about 25 minutes. Let cool to room temperature.
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2
Meanwhile, tear the bread into small pieces (about 1/2-inch squares). On another parchment-lined baking sheet, spread the breadcrumbs out, sprinkle with salt and pepper, and drizzle with olive oil until lightly coated. Toast in the oven until golden brown and crispy, about 15 minutes.
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3
Peel and core the apple, then cut it into small (say, 1/2 inch) cubes. Cube the cheddar the same size, too. In a small bowl, whisk together the apple cider vinegar, lemon, honey, and a pinch of salt and pepper. Then whisk in about 3/4 cup olive oil-enough so that the dressing is emulsified and tastes good to you.
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4
Once the brussels sprouts are at room temperature, add them to a mixing bowl with the apple, cheddar, and arugula. Pour over a little vinaigrette (you may not need all of it), gently toss, and add more vinaigrette if the salad looks dry. Then add the breadcrumbs and serve.
