This elegant little appetizer is savory and wonderful. Plus, it's make-ahead friendly and so much easier than it looks. You'll need a mini-muffin pan.
Preheat the oven to 425° F. Brush a mini-muffin pan with oil or melted butter. Crack the egg into a medium bowl and beat lightly. Peel the potatoes and cut with the julienne blade of a mandoline. (Hold the potato so as to cut crosswise, not lengthwise, otherwise the pieces will be too long.) Mix the cut potatoes with the egg as you go so they don't oxidize. Mix in a good pinch of salt and some freshly ground pepper.
Distribute the potato mixture among 24 mini-muffin molds to make a thinnish layer on the bottom and up the sides of each mold. Be careful to leave behind or drain off any accumulated liquid in the bowl rather than adding excess liquid to the molds. Bake for 25 minutes or until lightly browned and slightly crisp.
Cool on a rack for 15 minutes before removing nests from pan, loosening each by running a paring knife carefully around the sides.
While the potato nests bake, melt the butter and olive oil in a wide frying pan over medium-high heat. Add the minced shallot and garlic. Sauté for one minute. Add the sliced mushrooms, along with a generous pinch or two of salt and some freshly ground black pepper. Sauté, stirring occasionally, until the mushrooms release any liquid and it cooks off. Add the Marsala and cook, stirring once or twice, until all liquid has evaporated.
Turn off the heat and stir in the crème fraîche, grated cheese and tarragon.
Spoon a small amount of the mushroom mixture into each potato nest. Rewarm in a low oven for a few minutes before serving, if desired.