For the mashed potatoes: Put the potatoes in a large pot with 2 gallons of cold water. Bring the water and potatoes to a boil and cook until the potatoes are tender. Strain the potatoes and allow them to rest in the strainer over the sink for 5 to 10 minutes so they steam and dry out a bit. While the potatoes are draining combine the cream and butter in the pot and bring to a boil, melting the butter.
Put the potatoes back into the pot with the cream and the butter and whip with an electric hand mixer until the potatoes are smooth. Season with salt and pepper to taste. Keep warm. (If the potatoes cool off, heat them in a pan with the additional 1/4 cup cream and 2 tablespoons butter just before plating.)
For the vegetables: Prepare a large bowl of ice water for shocking the vegetables after blanching. Bring 2 gallons of water to a rolling boil and add the salt.
Add half of the green beans to the boiling water and cook until tender but not over-cooked, about 3 minutes. Transfer the beans to the ice bath using a hand strainer. Allow the water to come back up to a rolling boil and repeat the process with the remainder of the green beans and then the carrots.
For the trout: Line a baking sheet with paper towels. Heat a large skillet over high heat until smoking hot and reduce the heat to medium-high. Season the trout with salt and pepper, then dredge the skin side of the trout in the flour. Coat the skillet with grape seed oil.
Place the trout in the skillet skin-side down and sear until the skin is golden brown and the flesh begins to turn opaque. Gently flip the filets over and finish cooking for an additional 2 minutes. Transfer the trout to the prepared baking sheet and allow the cooked trout to rest for 5 minutes.
For the sauce: Meanwhile, preheat the oven to 350 degrees F. Spread the almonds on a baking sheet and toast in the oven until they just begin to brown, about 5 minutes. Set aside.
Heat a large saute pan over high heat until it begins to smoke. Add the grape seed oil to the pan and then add the shallots and capers. Saute for 2 minutes.
Add the lemon juice and bring to a boil; cook until reduced by half. Take the pan off the heat and add the butter a little at a time, swirling the pan to emulsify the butter into the lemon juice. Season the sauce to taste with salt and pepper (be careful with the salt because the capers can be very salty already). Stir in the chopped parsley.
For plating: In a large saute pan over high heat add the carrots and green beans and enough water to create a thin layer in the pan. Allow the water to boil and once the beans and carrots are hot, season them with salt and pepper.
Place a scoop of the mashed potatoes in the center of each of 6 plates. Arrange two filets of trout on top, flesh-side up. Arrange the carrots and green beans leaning on the trout to one side.
Spoon 2 ounces of sauce onto each plate and sprinkle the toasted almond slices over the top of the fish. Serve immediately.