Preheat the oven to 425°. In a large bowl, toss the brussels sprouts with 1/4 cup of the olive oil and season with salt and pepper. Spread the brussels sprouts on 2 large rimmed baking sheets and roast for about 45 minutes, stirring once or twice, until tender and charred in spots; shift the pans halfway through roasting.
Spread the walnuts in a pie plate and toast for about 8 minutes, until golden. Let cool, then coarsely chop the nuts.
In the large bowl, whisk the vinegar with the mustard and honey. Whisk in the remaining 1/2 cup of oil until emulsified. Add the capers, garlic, shallots and anchovies; season with salt and pepper. Add the brussels sprouts and walnuts and toss well. Serve.
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