Preheat your oven to 400°F (200°C).

In a food processor, finely chop the mushrooms until they are minced. Scrape down the sides as needed for an even consistency.

Pat the chicken breasts dry with a paper towel and slice each breast in half to create two pieces. Season both sides with kosher salt and black pepper.

In a large non-stick or cast iron pan, heat 1 teaspoon of olive oil and 1 tablespoon of butter over medium-high heat. Sear the chicken breasts for about 2 minutes on each side until golden brown but not fully cooked. Remove from the pan and set aside on a plate.

Add another teaspoon of olive oil and 1 tablespoon of butter to the same pan. Add the minced mushrooms, shallot, garlic, thyme, and season with 1/2 teaspoon of kosher salt and 1/4 teaspoon of black pepper. Cook, stirring often, for about 6-8 minutes until browned and no liquid remains. Transfer the mixture to a bowl and let it cool completely.

Lightly flour a clean surface and roll out the puff pastry into an 11-inch square. Cut the pastry into 2 squares. Place the squares on a baking sheet lined with parchment paper, leaving about 2 inches between them.

Spread about 1/4 cup of the mushroom mixture on each puff pastry square, leaving about an inch around the edges. Place a chicken breast on top of each, then spread about 1 teaspoon of Dijon mustard over the chicken.

Fold one edge of the puff pastry over the chicken, then fold the opposite edge over it. Bring the two ends together to form a tight pocket, sealing any openings. You can use toothpicks to secure the edges if needed. Brush the tops and sides with the beaten egg.

Bake in the preheated oven for 17-20 minutes, or until the puff pastry is golden brown and crispy, and the chicken reaches an internal temperature of 165°F (74°C). Cooking times may vary based on the thickness of the chicken breasts.

While the Chicken Wellington is baking, prepare the Dijon cream sauce. In a medium saucepan, melt 1 tablespoon of butter over medium heat. Add the minced shallot and sauté for 1 minute until fragrant.

Pour in the vegetable broth and increase the heat. Simmer until the broth is reduced by half (about 1/2 cup remaining). Stir in the heavy cream and continue to simmer until thickened, stirring occasionally. The sauce should leave a line when a wooden spoon is run through it. Stir in the Dijon mustard and season with salt and pepper to taste. The sauce will thicken as it cools.

Serve the Chicken Wellington hot with the Dijon cream sauce drizzled over the top. Enjoy!