Clean the mushrooms with a damp paper towel. Remove the stems and set aside.
Melt 2 tablespoons of the butter over low heat in a small saucepan or skillet. Dip each mushroom cap in the melted butter and place top side down in a baking pan.
Finely chop the mushrooms stems. Measure 1/4 cup of chopped stems and sauté with the chopped green onions and pecans in the remaining 2 tablespoons of butter until onions are wilted. Add more butter if needed.
Add the wine (or water) to the sautéed onions and mushrooms. Stir in the stuffing mix. Add additional wine or water a tablespoon at a time if needed to make a mixture that holds together when lightly pressed.
Spoon the mixture into the prepared mushroom caps.