Heat a medium skillet over medium heat. Add the walnut pieces and cook, stirring occasionally, until evenly toasted, 6 to 8 minutes. Remove to a small bowl.
Bring a medium pot of salted water to a boil.
Cut the broccoli florets into bite-sized pieces and thinly slice the stems. Have a bowl of ice water ready. Cook the broccoli in the boiling water until crisp-tender, about 3 minutes. Add the carrots and cook 30 seconds more. Drain the broccoli and carrots and submerge in the ice water to stop the cooking. Drain well and pat dry.
Whisk together the vinegar, honey and Dijon in a medium bowl with 1/2 teaspoon salt and pepper to taste. Gradually add the olive oil while whisking constantly. Add the broccoli, carrots and walnuts and toss to coat.
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