Heat the olive oil in a skillet over medium heat. Add the mushrooms and onion. Cook for 8 to 10 minutes, stirring often, until softened and most of their moisture has evaporated. Set aside to cool.
Heat the oven to 400°F. In a mixing bowl, combine the ground turkey, cooled mushroom mixture, egg, Dijon mustard, thyme, salt, and black pepper. Mix gently until evenly combined.
Shape the turkey mixture into a compact loaf about 8 inches long. Place it on a parchment-lined baking sheet and bake for 20 minutes. Remove it from the oven and let it cool for 10 minutes. Keep the oven on.
Lay the thawed gluten-free puff pastry on a clean surface. Place the partially baked loaf in the center. Fold the pastry over the loaf, trimming excess pastry if needed, and press the seams together to seal. Place it seam-side down on the baking sheet.
Brush the pastry all over with the beaten egg. Cut a few small slits in the top to let steam escape. Bake for 25 to 30 minutes, until the pastry is golden and the turkey reaches 165°F in the center.
Rest the Wellington for 10 minutes before slicing. Use a sharp knife to cut it into thick portions.