Heat the oven to 400°F. Line a rimmed baking sheet with parchment paper.
In a mixing bowl, combine the ground turkey, stuffing mix, egg, milk, diced onion, Dijon mustard, and black pepper. Mix just until combined, then shape the mixture into a compact loaf about 8 inches long.
Lightly flour a work surface and roll the puff pastry into a rectangle large enough to wrap around the loaf. Set the turkey loaf in the center. Fold the pastry over the loaf, sealing the edges underneath. Place it seam-side down on one half of the prepared baking sheet.
Toss the green beans with the olive oil and salt. Spread them on the other half of the baking sheet, leaving space around the pastry. Brush the exposed pastry with the water.
Bake for 35 to 45 minutes, until the pastry is deeply golden and the center of the turkey loaf reaches 165°F. If the green beans are tender before the pastry is ready, transfer them to a serving dish and keep baking the Wellington.
Let the Wellington rest for 10 minutes before slicing. Serve the slices with the roasted green beans.