Cook the potatoes: Preheat the oven to 350 degrees F. Place the potatoes in a roasting pan and drizzle with olive oil. Season with salt and stir to evenly coat the potatoes. Roast for 30 to 35 minutes, until the potatoes are soft. Remove from the oven and allow to cool slightly. While the potatoes are still warm, gently press down on each one with the back of a pan or the side of a knife so it flattens to about 3/4 inch but stays intact (it's okay if the edges crack a bit, as they will get extra crisp when fried). Heat about 4 inches of canola oil in a heavy pot over medium heat. Once the oil reaches 350 degrees F, fry the crushed potatoes, in batches, for 5 to 6 minutes, until they become golden and crunchy. Drain on paper towels and season with salt.
Make the vodka creme fraiche: Combine the creme fraiche and vodka in a mixing bowl and whisk vigorously to whip and combine. Season lightly with salt. Refrigerate until ready to use.
Prepare the toppings: Carefully flake the trout fillets into small bite-size pieces. Fry the shallots: Separate the shallot slices into rings. Lightly dredge the rings in the flour. In a saute pan, heat 1/2 inch of olive oil to 350 degrees F. Fry the shallots for 2 to 3 minutes, until golden brown. Drain on paper towels and immediately season with kosher salt. To assemble, place a small dollop of vodka creme fraiche on each piece of potato and top it with a piece of trout. Garnish with the radish sprouts, chives and fried shallots. Season with fleur de sel and drizzle with a little extra-virgin olive oil.