Peel potato and slice on a Japanese mandolin with the fine teeth. Place sliced potatoes into water. Working in batches, fry potatoes at 325 degrees until golden. Place potatoes on a paper towel-lined plate to absorb excess oil. Season with salt and reserve.
Set up an immersion circulator at 64 degrees Celsius. Place eggs in water bath and cook for 45 minutes. If you do not have a circulator, simply poach the eggs the classic way.
In a small saucepot, combine wine, sliced shallots, and thyme and reduce over low heat until wine has almost completely evaporated. Add cream, and then whisk butter a few cubes at a time until incorporated. Strain and add trout and chives. 4. Place one egg per bowl and ladle the trout and butter mixture around the egg, then garnish with crispy potatoes.