Pound the chicken between sheets of waxed paper using a flat meat mallet. You want to even out the thickness and thin them out a bit. Pat the chicken dry.
Combine the cheese, tomatoes, pine nuts, onions and basil in a small bowl. Season with salt and pepper. Spread the cheese mixture lengthwise over half of each chicken breast. Tuck the ends in and roll the chicken up tightly, securing with toothpicks. Dip the chicken in egg, and then roll it in breadcrumbs.
Place the chicken in an 8-inch square baking dish. Pour 2 tablespoons melted butter over the breasts. Bake until cooked through, about 20 minutes.
PREPARE THE SAUCE:
While the chicken is baking, melt 3 tablespoons of butter in a large skillet over medium heat. Add the mushrooms and sauté until tender (about 6 minutes). Add the wine and boil for 3 minutes. Add the chicken broth and boil until the liquid is reduced by half, about 5 minutes. Remove from heat and swirl in 4 tablespoons of butter, 1 piece at a time. Season with salt and pepper.
Remove the toothpicks from the chicken. Cut the chicken rolls into ½ inch thick rounds and serve them immediately with the sauce drizzled on top.
Notes
The chicken-rolling portion of the recipe may be prepared a few hours ahead of time and refrigerated.