Add the trimmed chicken breast pieces to the bowl of a food processor and puree. Transfer the minced chicken to a medium bowl and set aside. You do not need to clean out the bowl o the food processor, as it will be used to puree the vegetable mixture next.
Heat 1 tablespoon oil in a skillet over medium-high heat until shimmering. Add the mushrooms and cook, stirring occasionally, until the mushrooms are golden and the liquid has evaporated, about 10 minutes.
Add 1 tablespoon more oil to the pan with the leek and cook until softened, stirring frequently, about 3 minutes.
Add the garlic and thyme and cook about 30 seconds. Add 1 1/2 teaspoons lemon juice and cook for 30 seconds until the liquid has evaporated.
Transfer the vegetables to the bowl of a food processor, and return the pan to the heat.
Add the wine and scrape the pan to loosen browned bits,. Transfer the wine to a small bowl and set aside. Rinse and dry the skillet.
Process the vegetables in the food processor, then transfer to the bowl containing the minced chicken. Add 1 1/2 teaspoons parsley, 3/4 teaspoon salt, and 1/2 teaspoon pepper. Fold together with a silicone spatula until well combined.