Add the olive oil and butter to a skillet and heat over medium heat. Add the carrots, celery, onion and mushrooms, stirring occasionally until cooked through. Stir in the garlic for the last minute. Remove the vegetables and keep warm.
Add water to a pot and bring to a boil. Add in the broccolini, cauliflower, and sweetpotatoes. Boil for about 7-8 minutes or until cooked through. Drain in a colander.
In the skillet, add in the 3T butter and heat until melted. Using a whisk, stir in the flour and cook for about 30 seconds, until browned. Pour in the wine and stir to combine. Next, whisk in the stock and cream. Add one or two grinds of salt and pepper. Throw in the fresh chopped tarragon and simmer for about 10 minutes until thickened.
Add in all the cooked veggies to the sauce and stir to combine. Remove from the heat.
Cut the puffed pastry into triangle shapes and place them on top of the vegetable ingredients, making sure to overlap to cover. Tuck in a few larger pieces of tarragon. Brush with the beaten egg.
Place the pot pie into the oven and bake for 35 minutes or until golden and bubbly.
Notes
To prep in advance, prepare the filling and refrigerate (can be made up to 2 days in advance). When ready to serve, top with puff pastry and bake.