Preheat oven to 400°F (200°C). Line a baking sheet with parchment paper.
In a skillet, heat the olive oil over medium heat. Add the onion and garlic, sautéing until the onion is translucent and the garlic is fragrant.
Add the spinach to the skillet and cook until wilted. Remove from heat and let it cool slightly.
In a bowl, mix the cooled spinach mixture with cream cheese and dill. Season with salt and pepper.
Pat the salmon fillets dry with paper towels and season them with salt and pepper.
On a lightly floured surface, roll out the puff pastry sheet. Cut into four equal squares.
Spread a quarter of the spinach and cream cheese mixture in the center of each puff pastry square. Place a salmon fillet on top of each. Fold the pastry over the salmon, trimming any excess, and press to seal the edges. Repeat with remaining ingredients.
Using a sharp knife, gently score crosshatch slits on the top of each pastry-wrapped salmon. Be careful not to cut through the pastry.
Brush the beaten egg over the top of each Wellington.
Place the Wellingtons on the prepared baking sheet. Bake for 20-25 minutes, or until the pastry is golden and puffed and the internal temperature of the salmon reaches 145°F (63°C).
Let the Salmon Wellington rest for a few minutes before serving. Serve with lemon wedges.