Special equipment: 11-inch tart pan with a removable bottom parchment paper pie weights or dried beans
For the dough: Add flour and salt to the bowl of a food processor and pulse a few times to mix. Add the butter to the bowl and pulse until the mixture looks like crumbly sand. Add the egg and water and pulse again until the dough comes together.
Very lightly flour the work surface. Dump the dough onto the floured surface and knead a few times until it comes together. Shape the dough into a disc and roll out to a 13-inch circle. Press the dough into the bottom and sides of an 11-inch tart pan with a removable bottom. Level the edges with your fingers. Prick the bottom of the tart with a fork. Freeze until firm, 25 minutes.
For the filling: Preheat the oven to 400 degrees F.
While the tart is in the freezer, add the potatoes to a medium saucepan and cover with cold water. Add a big pinch of salt and bring to a boil. Reduce the heat to a simmer, and cook the potatoes until tender but still with a bit of resistance, 5 to 6 minutes. Drain and cool.
Lightly drizzle the olive oil in a large skillet over medium heat. Add the sausage, and cook, breaking up with the back of a wooden spoon until crumbly and browned, 6 to 8 minutes. Add the spinach by the handful and stir until it just wilts down. Season the mixture with salt and pepper and stir in the green onions. Remove to a bowl to cool.
Crack the eggs into a medium bowl and stir in the milk. Whisk together until combined and then add some salt and pepper.
Remove the tart from the freezer. Cover the top with parchment paper and add pie weights or dried beans. Place on a sheet tray and bake for 20 minutes.
Remove the pie weights and parchment paper and continue baking for 10 minutes more.
Turn the oven temperature down to 350 degrees F.
Evenly sprinkle the par-baked tart shell with half of the grated cheese, then the cooked potatoes, and finally top with the sausage and spinach mixture. Pour the eggs over the top and sprinkle with the remaining cheese. Bake for 45 minutes.
Cool the tart for 10 minutes before serving. Can be served warm or at room temperature.