Wrap the garlic cloves together in a piece of parchment paper; wrap each beet separately in parchment. Place the garlic and beetroots in a microwave-safe dish and cook for 3 minutes. Remove the garlic and set aside, still wrapped. Continue to cook the beetroots in intervals of 2 minutes, until they can be easily pierced with a knife, 8 to 10 minutes. Let sit wrapped in parchment to complete the cooking process and until cool enough to handle, about 15 minutes.
Meanwhile, whisk together the yoghurt, broth, vinegar, oil and 1/8 teaspoon salt in a small bowl. Peel the garlic cloves, mash completely and whisk into the vinaigrette. Season with salt and pepper. (Alternatively, make the vinaigrette with the microwave-roasted garlic up to a few hours in advance to allow the flavors to blend. Cover and refrigerate. Bring to room temperature before serving.)
Peel the beetroots using a paring knife and cut into wedges or slices. Arrange on a platter, drizzle with the vinaigrette and sprinkle with the mint and pistachios. Serve immediately, while still warm.