Preheat the oven’s broiler and oil a rimmed baking sheet with 1 tablespoon avocado oil.
Dry the flank steak well and season with salt and pepper. Place onto the baking sheet, rubbing the second tablespoon of oil all over the surface. Broil on high for 8-10 minutes, for medium rare (125ºF), or until cooked to your liking. Let the steak rest for 10 minutes before slicing. While the steak is cooking, prepare the sauce.
In a medium size pan, melt a tablespoon of butter and when hot, add the shallots. Cook, stirring occasionally until the shallots are softened and just beginning to caramelize, about 5 minutes. Add the garlic and cook until fragrant, one minute more. Deglaze with white wine, if using, and reduce until little to no liquid remains, about 5-10 minutes. Add the chicken broth and cook, until reduced by half, another 5-10 minutes. Reduce the heat to a bare simmer and add in the thyme and dijon. While whisking constantly, slowly incorporate the remaining butter cubes. A thickened and creamy sauce will be produced as the butter is emulsified. Stir in the tarragon and season to taste with more salt and pepper.
Slice steak against the grain (to ensure a more tender texture) and spoon sauce over top. Serve with a bit more fresh tarragon, if desired.