Cover the chicken breast with plastic wrap and gently pound to a uniform thickness to ensure even cooking.
Season the chicken breast with salt, pepper, and dried tarragon.
Pan saute the chicken breast in 2 teaspoons butter and 1 tablespoon olive oil over medium-high heat. Once the chicken in browned on one side, about 4 minutes, flip and add white wine and continue cooking until the internal temperature reads 165 degrees F on an instant read thermometer, about 5 minutes more.
In another medium skillet, heat the remaining 1 tablespoon olive oil over medium-high heat. Add the sliced mushrooms and cook until golden on each side and tender, about 5 minutes. Remove from pan. Add the spinach to the skillet and cook until slightly wilted, about 2 minutes.
Meanwhile, roast the red bell pepper under the broiler until the skin is blackened on all sides, about 6 minutes. Place the pepper in a bowl covered with plastic wrap and allow to steam 5 minutes. Peel, seed and slice into 1/4-inch strips.
In a small bowl, whisk to combine the basil pesto and mayonnaise.
Heat a large nonstick skillet or griddle on medium heat. Butter one side of both slices of the bread and place them butter-side down on a cutting board. Build each sandwich by beginning the layering on one of the slices of bread with the sliced mozzarella cheese, the sauteed mushrooms, spinach, roasted red peppers, and chicken. Spread 2 tablespoons of the pesto mayonnaise on the other slice and place it buttered-side up on top.
Place the sandwich on the griddle and cook until golden, about 4 minutes. Flip and continue to cook the other side until it is golden and the filling is warmed and cheese is melted, about 4 more minutes.
Cook's Note: Pan-sauteing the seasoned chicken breast then finishing it in white wine makes this sandwich a standout.