Preheat oven to 425 degrees. On a rimmed baking sheet, toss mushrooms with 2 tablespoons oil, and season with salt. Spread in a single layer, and roast 10 minutes. Toss in the scallions, and roast 15 minutes more. Push mushrooms and scallions to one side, and place spinach on empty side; roast until wilted, about 3 minutes. Let cool briefly; squeeze spinach dry.
Lightly brush an 11-by-7-inch rectangular tart pan with a removable bottom (or a 10-inch round pan) with oil. Working with one piece of phyllo at a time and keeping the rest covered with plastic wrap, brush a sheet of the pastry very lightly with oil. Fit into tart pan, leaving a 1-inch overhang. Brush a second sheet with oil, and fit into pan. Repeat with remaining sheets. Fold edges at top to make a border.
Crumple a double layer of foil into a rectangle the size of the bottom of the tart, and fit into crust to weigh down center. Place on a baking sheet, and bake until edges are golden and begin to set, about 7 minutes. Remove foil, and bake until golden all over, about 3 minutes more. (Tent edges with foil if browning too quickly.) Remove tart pan from oven.
Reduce oven to 375 degrees. In a blender, puree goat cheese, eggs, and milk until smooth; season with salt. Add herbs, and pulse to combine. Spread vegetables over crust, and pour custard over top.
Return tart pan to oven, and bake until custard is set, 20 to 23 minutes. Remove tart pan from oven, and let cool 10 minutes on a wire rack. Remove sides of pan, and let tart cool at least 10 minutes more. If not serving right away, slide tart off bottom of pan and directly onto rack. Serve warm or at room temperature.