Preheat oven to 450ºF (230ºC).
Melt butter in a small sauté pan. Set aside.
Using the slicer attachment to a food processor or a mandoline, slice the potatoes thinly. Transfer sliced potatoes to a large bowl of cold water. After letting them soak briefly, drain in a colander and then pat dry with kitchen towels.
On a baking sheet, brush the bottom and sides of two 4" non-stick round spring form pans, or ring moulds, with butter. Arrange the potato slices in a circle to line the pans, beginning from the outside in to the centre. Brush lightly with butter and season with salt and pepper.
Cover with a round of buttered aluminium foil and bake in the centre of a preheated oven for 20 minutes. Remove the foil and bake for an additional 10 minutes, or until the potatoes are cooked through and the edges are lightly browned.
Meanwhile, in a small bowl, combine together the herbs and Parmesan cheese. Set aside.
In the same sauté pan used for the melted butter, sweat the shallot and garlic over medium heat until translucent, about 10-12 minutes.
Divide the shallot mixture between the two pans. Carefully crack two eggs into each mould (you may want to use a separate small bowl to crack the eggs into first, then transfer them to ensure you do not break the yolks). Sprinkle over the herb and cheese mixture, spoon one teaspoon of cream over each and season well with salt and pepper.
Return to the oven and bake for 12-15 minutes, until the whites are cooked and just set. The eggs will continue to cook after you take them out of the oven, so take this into account when looking for doneness.
Allow the eggs to sit for approximately 1 minute. Use the tip of a butter knife to loosen the tart from the sides of the pan. Release the spring and use a wide, offset spatula to remove from bottom. If using a ring mould lift entire tart (with ring) to serving plate with spatula, then remove the ring.
Serves 2, best eaten immediately