1. Preheat oven to 200°C. Place mushroom, shallot and tarragon in a food processor; pulse to combine.
2. Heat olive oil in a large frypan set over medium heat.
3. Add mushroom mixture; cook until moisture evaporates and mushroom is golden brown, about 6 minutes. Stir in goat’s cheese.
4. Spread mushroom mixture under the skin of each chicken breast. Roast on a baking tray until cooked through, about 25 minutes.