Process mushrooms in a food processor until minced very small, scraping down the sides of the bowl as needed.
Pat chicken breasts dry with a paper towel and slice them in half, creating two separate (roughly) 4 inch breasts. Season both sides lightly with kosher salt and black pepper. Heat 1 teaspoon of olive oil and 1 tablespoon of butter in a large non-stick or cast iron pan over medium high heat. Sear chicken breasts about 2 minutes per side, until golden brown but not cooked through. Remove and set aside on a plate.
Add another teaspoon of olive oil and tablespoon of butter to the pan, then add minced mushrooms, shallot, garlic, thyme, and about ½ teaspoon of kosher salt and ¼ teaspoon black pepper. Cook, stirring often until browned and no liquid remains - about 6-8 minutes total. Transfer the mixture to a bowl and let it cool completely.
Lightly flour a clean work surface or cutting board and roll out the puff pastry into an 11 inch, even square. Cut the pastry into 2 squares. Place two of the squares onto a baking sheet covered with parchment paper, about 2 inches apart.
Spread about ¼ cup of the mushroom mixture on the top of the puff pastry, spreading out but leaving about an inch around the edges. Place the chicken breast on top, then add about 1 teaspoon of dijon to the top of the chicken, spreading it to coat the whole top. Fold one edge of the puff pastry in (see pictures in post for clear images), then fold the other corner over it. Take the two ends and place them over each other. This should form a tight pastry/pocket shape with no holes. You can secure also this with toothpicks to make sure it does not come apart.
Preheat the oven to 400 degrees Fahrenheit. Place the sheet pan, uncovered, in the fridge for at least 15-30 minutes, but not more. Just enough for the puff pastry to cool. When they have cooled, brush the tops and sides of the pastry with egg wash.
Bake at 400F for 17-20 minutes, or until the puff pastry has become golden brown and crispy, and the chicken has cooked through to 165 degrees F. Cooking times will vary depending on how thick your chicken breasts are. Mine were about ¾ inch thick and took 17 minutes.
While the chicken wellington is baking, make the dijon cream sauce. Melt butter in a medium saucepan over medium heat and add shallot, stirring for 1 minute. Add in vegetable broth and turn up the heat, simmer till it's reduced by about ½ (so ½ cup of broth remaining) then stir in the heavy cream. Continue to simmer till thickened - a wood spoon should leave behind a line when run through the sauce. Stir in dijon and season with kosher salt and black pepper to taste. The sauce will thicken as it cools.
Serve immediately and enjoy!