Heat oven to 400ºF.
Bring a small pot of water to a boil. Add the Swiss chard and cook until wilted, 2-3 minutes. Drain well.
Heat the olive oil in a large skillet. Add the onion and cook until softened, about 5 minutes. Add the peppers and mushrooms and cook another 5 minutes.
In a small bowl, whisk together the eggs and milk. Season with tarragon, salt, and pepper.
Arrange the Swiss chard in an even layer in the bottom of the pie crust. Top with cooked vegetables. Pour the egg mixture over the top, allowing it to seep down though the fillings. Dot with crumbled goat cheese.
Bake for 15 minutes. Reduce heat to 350ºF and bake another 20-30 minutes or until the center of the quiche is set.