Bring a medium pot of salted water to a boil. Add the green beans, and cook until bright green and tender-crisp, 2 to 3 minutes. Strain the beans in a colander, and run cold water over them until cooled. Season with 1/4 teaspoon kosher salt and a few grinds of pepper; set aside.
Heat the oil and butter in a large skillet over medium heat. Add the shallots, and cook, stirring frequently, until soft, about 3 minutes. Cut each Portobello cap in half, cut crosswise into 1/4-inch-thick slices and add to the skillet; season with 1/2 teaspoon kosher salt and a few grinds of pepper. Cook, stirring frequently, until the Portobellos have started to soften and wilt, about 4 minutes. Add the shiitakes, thyme, 1/2 teaspoon kosher 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 the liquid has evaporated, 2 to 3 minutes. Stir in the flour until the mushrooms are coated. Slowly stir in the half-and-half. Cook until thickened, about 3 minutes. Stir in the green beans, then remove the skillet from the heat and transfer the mixture to a large bowl. Let cool completely, then stir in 1 tablespoon of the breadcrumbs.
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 left and right sides of the pastry, pile the mushroom filling onto the puff pastry across the center, and shape 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. (You can use the parchment paper to help roll.) 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. Lightly sprinkle with sea salt and pepper. Bake until golden brown and bubbly, 30 to 35 minutes. Let rest for a few minutes.
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