Peel and dice the carrots, and cut the potatoes into 1/2 inch cubes. You don't need to peel the potatoes unless you want to, but be sure to scrub the potatoes if you don't peel them.
In a medium saucepan, combine the chicken broth, carrots and potatoes. Bring to a boil over medium heat, then boil for about 10-15 minutes, until the potatoes are tender enough to pierce with a fork. Set aside.
In a large saucepan, melt the butter over medium heat. Add the chopped onion, and cook for about 5 minutes, until the onion is softened. Add the garlic and cook for 1 more minute, until fragrant.
Stir in the flour until the onions and garlic are coated in flour, then stir in the white wine. The mixture will be thick and pasty.
Slowly stir in the milk, and stir until the mixture smooths out to a consistency like gravy.
Add the cooked chicken, salt, pepper, thyme, tarragon, and the chicken broth with the cooked potatoes and carrots. Bring the filling back to a boil over medium heat, and cook for 2-3 more minutes until slightly thickened. Remove from the heat and stir in the cream.
Place six 12-ounce ramekins on a baking sheet and divide the filling between the ramekins, or just pour the filling into a casserole dish. Cover the ramekins/casserole dish to keep the filling hot while you make the biscuits.