Pre-heat your oven to 400º Fahrenheit.
If necessary, begin by breaking down the lobster and retrieving all of the meat necessary. Read this guide for How to Break Down a Lobster *see notes*
Allow the puff pastry to thaw if you have not already.
Roughly dice 2 small shallots.
Peel and dice 2 yukon gold potatoes.
Take 1 large celery stalk and slice in in half longways.
Then, slice into small chunks.
Peel the skin off of 2 small carrots, then slice into half-moons. (Leave the skinny ends whole-moon to encourage even cooking).
Mince garlic and tarragon.
In a medium-large pot or dutch oven, heat 2 tablespoon olive oil over medium heat.
Add the shallots, carrots, and celery to the pot and stir.
Add potatoes and continue to cook until shallots become soft and translucent.
Next, add garlic and cook until it become fragrant and soft, just slightly browned.
Add the sherry cooking wine and stir to deglaze the pot.
Cook until wine is nearly evaporated entirely.
Melt 3 tablespoons of butter into the vegetable mixture.
Once melted, add the flour and immediately begin stirring to combine.
Cook for 1-2 minutes, until flour is slightly darker in color.
Whisk in broth until there are no remaining flour clumps.
Fold in fresh chopped tarragon and whisk in the cream.
Simmer until the filling has reduced and become thick (about 10 minutes).
Once thickened to your liking, turn off the heat and mix in the lobster meat.
Make an egg wash by whisking 1 egg together with a splash of water (water optional).
Pour the filling into an oval-shaped baking dish of 9 to 12 inches long.
Be sure to spread it flat with a rubber spatula and make sure it isn't too high that the puff pastry will stick to it.
Gently place a sheet of puff pastry over the top.
With your fingers, press the puff pastry to seal around the edges of the dish. Then, use a butter knife to cut off the excess pastry.
Take a paring knife and cut a small X shape in the center to vent.
Gently brush the egg wash over the top of the push pastry, being sure to cover all of it.
Bake at 400º F for 25-30 minutes, or until the puff pastry is evenly browned and "puffed" entirely.
Remove from oven and cool.
Serve warm and enjoy!