Heat 2 tablespoons oil in a large skillet or Dutch oven over medium-high heat.
Add the minced yellow onion and minced garlic and sauté until the onion is softened and begins to turn golden, about 3–5 minutes.
Add the minced cremini, white, and portobello mushrooms, minced rosemary, dried thyme, and a pinch of salt; stir to combine.
Cook the mixture over medium heat, stirring occasionally, until the mushrooms release their liquid and most of it has evaporated and the mushrooms are tender, about 15–20 minutes.
Remove the pan from heat and stir in the toasted minced walnuts, tamari (or soy sauce), balsamic vinegar, and freshly cracked black pepper; adjust salt to taste. Let the filling cool.
On a floured surface, roll one chilled dough ball slightly thinner than a pie crust. Use a 6-inch cookie cutter (or desired size) to cut rounds; repeat until all dough is cut.
Preheat the air fryer to 350°F (or preheat oven to 350°F if baking). Whisk together soy milk and maple syrup in a small bowl for brushing.
Place about 1 1/2–2 tablespoons of cooled mushroom filling in the center of each dough round. Fold the dough over to form a half-moon and press edges together, then seal with a fork.
Spray or brush the air fryer basket with oil. Arrange empanadas in a single layer with at least 1/2" space between them (cook in batches if needed).
Brush the tops of the empanadas with the soy milk–maple mixture. Air fry at 350°F for 10 minutes, then check for golden color; if needed, flip and air fry an additional 5 minutes.
Remove empanadas and transfer to a plate. Repeat with remaining empanadas until all are cooked; you should have about 20 empanadas total.