To make the crust: Place the flours, butter, and salt in a food processor and pulse until crumbly. Add as much of the ice water as necessary and continue to pulse to make the dough come together, taking care to not overwork it. Gather the dough into a disk and wrap it in plastic. Refrigerate for 1 hour.
Preheat the oven to 375 degrees F.
Roll the dough out on a lightly floured surface until it's a circle 12 to 13 inches in diameter. Transfer to an 11-inch tart pan with a removable bottom and press it in evenly. Line the dough with a sheet of parchment paper or foil, then fill with pie weights or dried beans to weigh down the dough as it bakes. Bake for 15 to 20 minutes, or until the crust begins to brown, then remove the weights, prick the bottom of the curst, and continue to bake for another 10 to 12 minutes, or until the bottom is golden.
Meanwhile, to make the filling, in a large skillet, melt the butter. Add the mushrooms and sauté over medium-high heat for 5 minutes, or until they've softened up a bit. Spoon the mushrooms evenly over the crust.
In a large mixing bowl, beat the eggs, then stir in the cheese and milk. Pour this mixture over the mushrooms in the crust. Bake until golden and set, about 25 minutes.
Let the quiche cool to room temperature to serve it the authentic Swedish way.