Preheat the oven to 350°F.
Bake the potatoes until they are soft inside, about 45 minutes to 1 hour. As soon as they are cool enough to handle, peel the potatoes-you should have about 3 cups-and place them in an electric mixer with a whip attachment, along with the salt and brown sugar. Whip the potatoes at medium speed and when the mixture is smooth, about 2 minutes, add the melted butter. Continue to whip until the butter is incorporated, then add the yogurt and the eggs. Beat at high speed until the color lightens and the mixture becomes light and fluffy, about 5 minutes.
Pour the potato mixture to a 9-inch X 13-inch casserole dish or baking pan, or to 6 individual ramekins, then shove bits of the Stilton cheese into the mix at random places. Place the pan or dishes in another pan large enough to hold them. Pour enough water into the larger pan to come about two-thirds the way up the sides of the casserole dish or ramekins, and bake until the cheese bits are melted and the top is just beginning to brown, about 45 minutes. Serve hot or let cool to room temperature, but do not serve it cold. If you refrigerate the pudding, warm it slightly before serving.