Put the potatoes in a large pot of cold water and bring to a boil. Reduce heat and simmer for 15-20 minutes or until potatoes are very tender. Drain well.
Rub a 9×13 baking dish with 2 tablespoons of butter, and then sprinkle with 1/4 cup Parmesan cheese, making sure the cheese is evenly distributed.
Put the potatoes in a large bowl (preferably the bowl of a stand mixer) and add the Greek yogurt, Chèvre, 2 tablespoons butter, 1/4 cup Parmesan, and the chives, if using. Season with salt and pepper.
Using an electric mixer (with a whisk attachment if you have one), beat the potatoes until smooth and light, being careful not to over mix as the starch in the potatoes can cause them to become gluey if mixed too long.
Transfer the potatoes to the prepared dish and dot with the remaining 2 tablespoons butter and sprinkle with the remaining 1/4 cup Parmesan.
At this point, you can cover tightly and refrigerate for 2 or 3 days or wrap well and freeze for up to a month.
Preheat oven to 350℉ and bake until the potatoes are golden and hot throughout, about 25 minutes. If the potatoes have been frozen or refrigerated, thaw and bring to room temperature before baking. If the potatoes are cold, extra time will be needed to heat through.