Preheat the oven to 425 degrees F. Line 2 baking sheets with parchment paper.
On one prepared baking sheet, toss the carrots with the olive oil, honey and some salt and pepper. Bake until fork-tender, 20 to 25 minutes. Remove from the oven and reduce the oven temperature to 400 degrees F.
Put the mushrooms in a food processor and pulse until finely chopped. In a medium skillet over medium heat, melt the butter. Add the garlic and shallots and saute until fragrant and the shallots are translucent, 3 to 4 minutes. Add the mushrooms and cook until all the water has been released and the mushrooms are browned, 8 to 10 minutes. Season with salt and pepper. Let cool completely. (This step can be done a day or two in advance.)
Sprinkle a clean, flat surface with flour. Use a floured rolling pin to roll out the puff pastry to approximately a 10-by-12-inch rectangle. Use a spatula to spread the mushroom mixture onto the pastry dough, leaving a 1/2-inch border. Place the carrots in an even row onto the mushroom mixture, alternating big and little ends to make the layer flat. Sprinkle with the goat cheese and parsley. Use a pastry brush to coat the border of the pastry with egg wash, then roll the pastry 3 times onto itself, into a cylinder. Tuck the ends of the pastry under and pinch the ends closed with your fingers. Place on the remaining prepared baking sheet. Use the pastry brush to lightly coat the wellington in egg wash, taking care not to miss any spots. Using a paring knife, cut 4 slits down the center of the pastry.
Bake until golden brown, 20 to 25 minutes. Let stand 5 minutes before slicing.