Drain the yogurt in a paper towel-lined strainer set over a bowl, until thick, about 1 hour. Discard the watery liquid. Whisk the horseradish and mayonnaise into the thickened yogurt.
Put the chicken in a saucepan along with the vermouth, garlic, salt, rosemary, bay leaf, red pepper flakes, and water to cover. Bring just to a boil then reduce to a simmer. Cook covered, just until firm, about 10 minutes. Pull pan off of the heat, and allow the chicken to cool in the poaching liquid. When chicken is cool, shred into bite-size pieces.
Whisk the mustard, lemon zest and juice, salt, and pepper, to taste, in a large serving bowl. Gradually whisk in the oil, starting with a few drops and then adding the rest in a steady stream, to make a smooth dressing. Toss the chicken, celery, and radishes, in the dressing. Season with salt and pepper, to taste. Drizzle with some of the horseradish cream.
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