Preheat oven to 375°F.
Place sliced potatoes into a mixing bowl and top with cream. Toss gently to coat. Season with salt and pepper. Allow potatoes to sit for 5 minutes, or longer if you can.
Line a baking sheet (with a lip) with parchment and begin placing the potatoes in a single layer, slightly overlapping one another. Sprinkle the surface with some shallots, garlic and cheese. Dot the surface with a small amount of butter and lightly season with salt and pepper.
Continue layering potatoes and shallot mixture until all the potatoes have been used, placing small dots of butter between each layer.
Top gratin with remaining butter and season bake for 15 about 15 minutes. Remove from oven and sprinkle with remaining cheese. Place back into the oven for another 10-15 minutes until the potatoes are fork tender and the cheese has bubbled and browned on top. Allow gratin to cool for 5 minutes. Cut into wedges and serve with béarnaise sauce.
Béarnaise sauce: Place 2 teaspoons tarragon, vinegar and shallot into a small saucepan and reduce by half. Strain and set aside until completely cooled.
Place egg yolks and vinegar reduction into a stainless steel or heatproof glass bowl and whisk together until the mixture doubles in volume. Place mixture over a pot with simmering water (make sure the water from the pot does not reach the bottom of your bowl) and begin whisking. While whisking, begin to add melted butter, in a slow and steady stream, until fully incorporated and the béarnaise is thick and smooth. (the sauce should gradually thicken as you add the butter) Remove from heat and add remaining tarragon and season with salt and freshly cracked black pepper.
Drizzle sauce over gratin wedges and serve.