Preheat your oven to 425°F.
Place the potatoes in a small pot of cold, salted water. Bring to a boil. Add the carrots to the potatoes once the water starts boiling. Cook until just fork tender.
In a large skillet over medium-high heat, melt the butter, then add the shallot and garlic. Cook 1-2 minutes, until just starting to soften.
Next add the chicken and cook for another 1-2 minutes to warm. Stir in the flour, salt, and tarragon.
Mix in the milk and cream and bring to a boil. Reduce heat and simmer until the sauce thickens.
Once the sauce is thick, mix in the wine, potatoes, carrots, and peas.
Transfer to one large baking dish and top with a sheet of the thawed puff pastry. Make a few slits in the puff pastry for steam to escape while cooking.
Place the dish on a baking sheet and bake for 30 minutes or until the puff pastry is golden brown. If you are using smaller, individual baking dishes, bake them for about 20 minutes, instead.