Preheat the oven to 400°F. Add the quartered mushrooms to a food processor and pulse until they are very finely minced, scraping down the sides as needed.
Pat the chicken breasts dry with paper towels. Slice them in half to create separate pieces. Season both sides lightly with kosher salt and freshly ground black pepper.
Heat 1 teaspoon of olive oil and 1 tablespoon of butter in a large nonstick or cast-iron skillet over medium-high heat. Sear the chicken for about 2 minutes per side, until golden brown but not cooked through. Transfer the chicken to a plate.
Add the remaining teaspoon of olive oil and tablespoon of butter to the skillet. Add the mushrooms, minced shallot, garlic, thyme, ½ teaspoon kosher salt, and ¼ teaspoon black pepper.
Cook the mushroom mixture, stirring frequently, for about 6–8 minutes. Continue cooking until the mushrooms are deeply browned and all excess liquid has evaporated. Transfer to a bowl and allow the mixture to cool completely.
Lightly flour a clean work surface. Roll the thawed puff pastry into an even 11-inch square, then cut it into 2 squares.
Place the pastry squares on a parchment-lined baking sheet, leaving about 2 inches between them.
Spread approximately ¼ cup of the cooled mushroom mixture over each pastry square, leaving about 1 inch of pastry uncovered around the edges.
Place a seared chicken piece over the mushroom mixture. Spread about 1 teaspoon of Dijon mustard over the top of the chicken.
Fold the pastry around the chicken, overlapping the edges to create a snug, enclosed pocket. Make sure there are no open holes or gaps. If needed, secure the pastry with toothpicks.
Brush the tops and sides generously with the beaten egg.
Bake the chicken puff pastry at 400°F for 17–20 minutes, or until the pastry is golden brown and crisp and the chicken reaches an internal temperature of 165°F. Thicker chicken pieces may require additional cooking time.
While the Wellington bakes, prepare the Dijon cream sauce. Melt the butter in a medium saucepan over medium heat. Add the minced shallot and cook for about 1 minute.
Pour in the vegetable broth and increase the heat. Simmer until the broth has reduced by approximately half, leaving about ½ cup.
Stir in the heavy cream and continue simmering until the sauce thickens enough for a wooden spoon to leave a visible line through it.
Stir in the Dijon mustard and season with salt and black pepper to taste. The sauce will become slightly thicker as it cools.
Serve the Chicken Wellington immediately with warm Dijon cream sauce spooned over or alongside the pastry.