Arrange racks in upper and lower thirds of oven and heat oven to 400°F. Place walnuts on small rimmed baking sheet. On separate small rimmed baking sheet, toss shiitake caps with 1 tablespoon oil and 1/4 teaspoon salt and arrange gill sides up. Roast walnuts on upper rack and mushrooms on lower rack until nuts are toasted, 4 to 6 minutes. Flip shiitake caps and continue roasting until tender and beginning to brown, 12 to 14 minutes more. Let nuts cool 5 minutes and let shiitakes cool to room temperature.
Meanwhile, in 4 batches, pulse mixed mushrooms in food processor until finely chopped; transfer to medium bowl. Wipe out food processor and pulse toasted walnuts until finely chopped; transfer to large bowl.
Heat remaining 2 tablespoons oil in large Dutch oven on medium. Add shallots, season with 1/4 teaspoon each salt and pepper and cook, stirring occasionally, until translucent and just tender, 4 to 5 minutes.
Add chopped mushrooms, season with 3/4 teaspoon salt, and cook, stirring occasionally, until mushrooms have released their liquid, about 5 minutes. Continue cooking, stirring occasionally, until liquid has evaporated and mushrooms have browned, 15 to 20 minutes more.
Stir in garlic and thyme and cook, stirring, 2 min. Reduce heat to low. Add sherry and soy sauce and cook, stirring, 2 minutes. Transfer mushroom mixture to bowl with walnuts, add parsley, and fold to combine. Let mixture cool 5 minutes.
Place large piece of plastic wrap on work surface. Place half of mushroom mixture on top and shape with hands into 11- by 2 1/2-inch rectangle. Arrange shiitake caps on top, lengthwise along center. Top with remaining mushroom mixture, gently packing with hands into cohesive mass. Wrap log in plastic, twisting ends to secure. Refrigerate 15 minutes.
Line large rimmed baking sheet with parchment. On lightly floured surface, gently unfold puff pastry and roll to 13- by 10-inch rectangle, with one longer side facing you. Spread Dijon on puff pastry, leaving 1-inch border on all sides. Unwrap mushroom log and place lengthwise along edge of pastry closest to you. Roll puff pastry to completely cover mushroom log. Carefully transfer to prepared baking sheet, seam side down, fold ends like a present to cover and pinch to seal, then tuck ends under log.
Brush top and sides of pastry with soy milk, then use small sharp knife to score dough in decorative pattern (or make a few slits on top). Bake on lower rack until pastry is deep golden brown, 35 to 40 minutes. Let rest 15 minutes, then top with thyme sprigs if desired. Slice and serve while warm.