Preheat the oven to 350F/177C. Prepare a non-stick 8x8 baking dish with parchment paper and lightly butter the sides.
Use a mandolin to thinly slice the potatoes lengthwise into ⅛ inch thickness.
In a large bowl, toss the thinly sliced potato in the heavy cream.
Add a single layer of the potatoes into the baking dish. They should overlap slightly but not too much. Season the top with a little bit of salt and pepper.
Mix together the shredded cheese and chives, then sprinkle 1-2 tablespoons of it on top of the potato. Dot about ½ tablespoon of the butter over the top, then continue with the next layer. Keep layering until all of the ingredients are used up, making the last layer just seasoned potato and dotted butter.
Tightly wrap the pan in foil and bake for 35-45 minutes, or until the potatoes can easily be pierced with a sharp knife. Then place a pan with a heavy object on top of it and let it cool for 1-2 hours before placing it in the fridge to chill overnight, or at least 6 hours.
Then, gently life up the sides of the parchment paper to remove the pave from the pan. Place it on a cutting board and use a sharp knife to trim the sides into a flat square. Cut the pave into 1-inch by 1-inch pieces and set them aside.
Heat a medium non-stick skillet over medium-high heat. Add enough oil to the pan to completely cover the bottom and allow it to get hot before adding the first piece.
The pave should not stick to the pan, if it does, the oil isn't hot enough or there isn't enough oil. Fry the pave cut side down for 1 minute, then gently turn them over and fry for another minute.
Serve immediately with more fresh chives!