For the dough, combine the dry ingredients in the bowl of a food processor. Pulse to mix. Cut butter into 12 pieces and add; pulse to mix in finely. Add egg and yolk and pulse to form a ball -- if it resists, add a teaspoon of water and continue. Form the dough into a disk and wrap in plastic. Refrigerate until needed, up to 2 days. When ready to bake the tart, preheat oven to 350 degrees and set a rack in the lowest level. On a floured surface, roll the dough and use it to line a 9 to 10-inch tart pan. For the filling, melt the butter in a small saute pan and cook the scallions for a few minutes to wilt them. Scrape onto the dough and spread evenly. Cover with the grated cheese and walnuts.
Whisk the remaining ingredients together in a bowl and pour into the tart crust. Bake the tart for about 30 minutes, or until the filling is brown and puffed and the crust is baked through. Serve immediately or cool and serve later.