Step 1 Break the stems off the mushrooms and wipe them clean with a damp paper towel. Cut the stems into 3 or 4 slices. Set them aside. Wipe the mushrooms clean with the paper towel. Slice the mushroom caps into 1/4-inch slices and add them to the stems.
Step 2 Melt 2 tablespoons of butter in a skillet, tilting the skillet when the butter is melted so the butter coats and covers the bottom. Add the sliced mushrooms, lightly salt them and stir so they are cooked on both sides, about 4 minutes. Set them aside.
Step 3Heat the milk in a saucepan over medium heat so it is just hot; don't boil the milk. Set the milk aside.
Step 4Using a 1 1/2-to 2-quart saucepan, melt the remaining 4 tablespoons of butter over medium heat. Tilt the pan so the butter coats the bottom of the pan. Sprinkle the flour over the melted butter and stir constantly until the butter and flour are smoothly blended, 2 to 3 minutes. Slowly add the hot milk, stirring constantly until the sauce thickens, 4 to 5 minutes. Add salt and pepper to taste and continue to stir. Stir the sauce over moderate heat for 3 to 4 minutes, then taste for enough salt and pepper. Add the lemon juice or onion, if desired. Stir to mix well. Add the cooked mushrooms. Serve hot.