Par-bake the crust. Preheat the oven to 375ºF. While the oven is heating, put the pie crust in its tin on the counter and let it thaw slightly, so you can prick it all over with a fork. Line the pie shell with foil or parchment paper, then weight it with dried beans or pie weights. Bake the crust until slightly golden on the edges, about 15 to 20 minutes. Transfer to a cooling rack and remove weights, simply picking the foil or parchment at the corners.
Meanwhile, make the filling. Heat the oil in a large sauté pan over medium flame. Add shallot and cook, stirring frequently, for a minute or two. Add mushrooms and toss to coat with oil. Add tarragon and toss to coat. Cook for about 3 minutes, turning the mushrooms halfway through.
Add chopped spinach to pan, salt lightly and toss to combine every thing. Cover with a lid and cook for about 5 minutes, stirring once or twice. The mushrooms will release liquid, helping spinach cook down. Uncover pan and cook another 2 to 3 minutes, allowing excess liquid to cook off.
Beat eggs in a bowl, then add half & half and beat to combine.
Assemble and bake the quiche. Spread the Dijon mustard in the bottom of the crust (this is optional, but the mustard sharpens the cheese flavors without imparting its own taste). Put mushroom/spinach mixture into the crust and gently spread with a fork to evenly distribute it. Sprinkle cheese evenly over the filling. Pour the egg mixture over the top, using your fork to gently help it settle in around the filling.
Transfer to the middle rack of the oven and bake the quiche until it has puffed up and is set and beautifully browned, about 25 to 35 minutes. You can test with a sharp knife to make sure the middle is set.
Cool on a rack at least 10 minutes, then serve. You can also serve it room temperature, but I think warm is best. See Kitchen Notes about reheating leftovers.