Wrap the garlic cloves together in a piece of parchment paper; wrap each beet separately in parchment. Place the garlic and beets in a microwave-safe dish and cook for 3 minutes. Remove the garlic and set aside, still wrapped. Continue to cook the beets 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 yogurt, 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 beets 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.
The beets are "roasted" in the microwave, saving time and energy. The garlic is also "roasted" in the microwave, to mellow its bite.