Preheat the oven to 400°F.
Clean the radishes, remove the greens and stems, cut into quarters, and place on a parchment-lined baking sheet. Cut the butter into small pieces and dot over top of the radishes. Season lightly with salt and pepper.
Roast the radishes for 20 to 25 minutes, stirring once halfway through, until they're soft and browned. Remove from the oven and cool.
Transfer the radishes to the bowl of a food processor, and pulse until broken down into small pieces, but not puréed. In a medium bowl, mix together the radishes, yogurt, chives, and lemon zest, and season with salt and pepper.
Serve immediately, or store in a covered container in the refrigerator until ready to serve.