Season the chicken breasts with 1/4 teaspoon salt and 1/4 teaspoon black pepper. In a large skillet, heat 1 tablespoons of olive oil over medium high heat. Add the chicken and sauté for 3 to 4 minutes per side until cooked through. Chicken is done when a meat thermometer inserted in the center reaches 165 degrees F. Transfer to a cutting board to rest. When cool enough to handle, shred chicken into bite sized pieces.
To the same skillet, heat 1 tablespoon of olive oil then add the mushrooms. Season with 1/4 teaspoon salt and 1/8 teaspoon black pepper (you can eyeball this if desired). Sauté until mushrooms are golden brown, about 7-8 minutes. Transfer the mushrooms to a bowl. Heat the remaining tablespoon of oil over medium heat. Add the garlic and sauté about 30 seconds. Add the Swiss chard and sauté until wilted, about 3 minutes. Season with 1/4 teaspoon salt and 1/8 teaspoon and black pepper. Transfer the Swiss chard to a plate to cool.
In a small bowl, combine the ricotta and Parmesan cheese, remaining salt and black pepper. Roll the dough into a 14-inch round and place on a parchment lined baking sheet. (I do this on the counter between 2 pieces of parchment and then transfer the crust directly on the bottom piece of parchment.) Spoon the ricotta cheese mixture over the dough, leaving a 2-inch border. Evenly top with the Swiss chard, the shredded chicken, and then the sautéed mushrooms.
Preheat the oven to 400 degrees F. Fold the edges of the dough up and over the chicken filling, creating a 1-inch border. Brush the dough border with the beaten egg. (I sometimes brush a little over the the topping, too.) Bake until the crust is golden brown and the filling is heated through, about 30-35 minutes. (Check after 25 minutes, as all ovens vary.) Remove the galette from the oven and let cool for a few minutes before cutting into wedges and serving.
The galette may be enjoyed warm or at room temperature.