Heat the olive oil in a large skillet over medium-high heat. Meanwhile, pat dry the chicken thighs with paper towel.
Once the oil is shimmering, add the thighs to the skillet and brown for 7½ minutes on each side. After a couple minutes of cooking, turn down the heat to medium-low and cook until the meat reaches 165°F (74°C) in the thickest part of the thigh.
While the meat is cooking, slice the leeks, mince the garlic, and chop the tarragon. Measure out the butter, flour, wine, and milk, as you won't have time to measure them out later.
Once the meat has browned, remove it to a plate with a pair of tongs. Take the skillet off the heat and scrape out most of the browned bits with a metal spatula.
Return the skillet to medium-high heat, then add the leeks and cook for 3 minutes, until softened. Add the garlic and cook until fragrant, about 1 minute. Deglaze the pan with ⅓ cup of white wine, stirring until the brown fond is gone from the bottom of the pan. Add additional wine if needed. Dump the cooked veggies into a bowl and set aside.
Melt the butter in the same skillet over medium-high heat. Dump in the flour and whisk constantly for 2 minutes to cook the flour. Immediately dump in the wine and milk, then whisk constantly until smooth and thickened. Add the chopped fresh tarragon, salt, and pepper, then cook for 2 to 3 minutes and remove from the heat.
Chop the cooked meat into bite-sized pieces, then stir it and the cooked veggies into the sauce. (You'll probably have a little extra meat; just set that aside to eat later.) Spread the meat filling into a 9-inch pie plate and set aside to cool completely. You can cover and refrigerate the filling at this stage and bake it another day.