Bring seafood stock to a simmer over medium heat and season with Kosher salt. Add lobster tails, cover and let them simmer for 8-9 minutes until cooked through. Remove lobster tails from broth and immediately plunge in cold water to stop the cooking process. Reserve seafood stock for later.
Remove the cooked lobster from the shells, then roughly chop and reserve for later.
Preheat oven to 375°F.
Heat 2 tablespoons of olive oil over medium heat in a Braiser. Add chopped onion and fennel bulb, season with Kosher salt and sauté until translucent and starting to brown, about 12-14 minutes.
Add garlic and grate the zest from the lemon into the pot and stir until fragrant, about 1 minute. Add flour and cook for another 1-2 minutes until the flour starts to toast. Add white wine, scraping up any browned bits, and cook for another 1-2 minutes until the mixture has a thick paste-like consistency.
Add seafood stock and the juice from the lemon and stir until the mixture is smooth. Bring to a simmer and cook until the mixture is thickened, about 3-5 minutes. Remove from heat and stir in heavy cream, peas, pearl onions, fresh herbs and cooked lobster.
Using the braiser lid as a guide, roll out your puff pastry so it’s large enough, then spread over the top of your liquid, pressing down into the edges of the braiser. Trim off any excess pastry. Tip: you can also cut the puff pastry into a lattice shape and layer them on top.
Make an egg wash with the egg and 1 tablespoon of water, then brush all over the top of the pastry. Season the top with Kosher salt and bake for 40-45 minutes until the pastry is puffed and golden brown and the liquid is bubbling along the edges.
Remove and cover with lid to keep warm until serving.