For the Wellington: Remove the stems and gills of the Portobello mushrooms, and discard. Cut each mushroom cap in half, then cut the halves into 1/4-inch-thick slices.
Heat the oil and butter in a large skillet over medium heat. Add the shallots, and cook, stirring frequently, until soft, 2 to 3 minutes. Add the Portobello slices, 1/4 teaspoon salt and a few grinds pepper, and cook, stirring frequently, until the mushrooms have started to soften and wilt, about 4 minutes. Add the shiitakes, thyme, 1/4 teaspoon salt and a few grinds of pepper, and cook, stirring frequently, until the shiitakes have started to soften and wilt, about 3 minutes. Pour in the wine, bring to a simmer and stir until most of the liquid has evaporated, 2 to 3 minutes. Remove from the heat, and let cool completely. Stir in 1 tablespoon of the breadcrumbs. Taste and adjust the seasoning.
Position an oven rack in the center of the oven, and preheat to 400 degrees F. Roll the puff pastry out on a floured sheet of parchment paper to a 13-inch square. Brush the surface of the pastry with the mustard, and sprinkle with the remaining 1 tablespoon breadcrumbs.
Leaving a 1-inch border on the right and left sides of the pastry, pile the mushroom filling onto the center, and pack it into a log about 3 inches wide. Fold the bottom half of the pastry over the filling, then roll it up so the bundle sits seam-side down. If any filling falls out while rolling, just push it back inside. Pinch together the open ends to seal, and crimp with a fork. The log should be about 12 inches long. Transfer the Wellington, on the parchment, to a baking sheet. Brush generously all over with the egg. Cut decorative slits down the length of the Wellington. Sprinkle with sea salt and pepper. Bake until golden brown and bubbly, 40 to 45 minutes. Let rest for a few minutes.
For the carrot sauce: While the Wellington is baking, combine 2 cups water, carrots, garlic, bay leaf and 1/2 teaspoon salt in a small saucepan. Bring to a high simmer, and cook until the carrots are very tender, 10 to 12 minutes. Remove the bay leaf, and transfer all the liquid and solids to a blender. Blend until completely smooth. Add the cream cheese, vinegar and 1/2 teaspoon kosher salt and blend until smooth again. (The sauce can be made, cooled and refrigerated up to 1 day ahead; reheat before serving.) Transfer the sauce to a serving vessel.
For the salad: Toss the frisee with the oil and vinegar, a pinch of salt and a few grinds of pepper.
Arrange the Wellington and salad on a platter. Slice the Wellington into 6 even pieces, and serve with the salad and carrot sauce.
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