Preheat the oven to 425 degrees F. Put the fennel in a medium bowl, cover with ice water and set aside.
Put the beets in a 3-quart casserole dish and toss with the chicken broth, 1 tablespoon of the olive oil, 1/2 teaspoon salt and a few grinds of pepper. Cover the dish with foil and bake for 45 minutes. Remove the foil; if there is 1/4 inch or less of liquid, add 1/2 cup of warm water. Continue baking, uncovered, until the cooking liquid is reduced by half and the beets are fork-tender, 25 to 30 minutes.
Meanwhile, whisk the vinegar, butter, remaining 2 tablespoons oil and mustard in a measuring cup. Strain the fennel and toss with the vinegar mixture in a large bowl. Add the dressed fennel to the beets, toss with tongs and continue baking until the fennel is crisp-tender, about 15 minutes. Remove the dish from the oven, stir the beets and fennel to coat with the sauce and season with 1/4 teaspoon salt and a few grinds of pepper. Transfer to a serving bowl and garnish with the chives.
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