Preheat the oven to 425°.
In a large bowl, toss brussels sprouts with 1/4 cup of the olive oil and season with salt and pepper.
Spread the brussels sprouts on baking sheet and roast for about 45 minutes, stirring once or twice, until tender and charred.
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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