Preheat oven to 350°F.
Combine the goat cheese, green onion, basil, 1/2 cup breadcrumbs, one whisked egg, salt, and pepper in a bowl. Stir to mix. Set aside.
Cut kitchen twine into 6 strips long enough to tie around the chicken breasts.
Pound chicken breasts between two sheets of wax paper placed on a cutting board. You want the breasts to be about 1/4" thick. This doesn't have to be precise, you just want an even, consistent thickness when done. (I use a meat mallet or if you don't have one try a frying pan to pound your breasts.)
Place a large spoonful of filling across the pounded chicken, close to the top. Roll the chicken up and secure with twine. (If you don't have kitchen twine, use toothpicks to secure.)
Beat the second egg and place on rimmed plate. Pour remaining bread crumbs on another rimmed plate. Place both plates next to a baking sheet lined with foil.
Place stuffed chicken breasts one at a time in egg to coat. Next, roll in crumbs to coat. Repeat with remaining chicken breasts and place on baking sheet.
Top each breast with a slice of butter.
Bake for 35 - 40 minutes. Remove from oven and allow to cool to touch, about 5 minutes.
Meanwhile melt 1/2 stick of butter in sauté pan over medium heat and add mushrooms, cook stirring occasionally about 6 - 7 minutes.
Add wine and stir. Allow to cook for 3 minutes, and then add broth.
Allow liquid to reduce by half. This should take another 6-7 minutes.
Remove pan from heat and add remaining butter one tablespoon at a time, stirring each tablespoon until combined.
Cut twine and remove from chicken. Slice and serve topped with mushrooms.