Preheat oven to 425 degrees.
On a baking sheet, toss mushrooms with 2 tablespoons oil and 3/4 teaspoon salt. Roast for 10 minutes. Toss in the scallions
and roast for 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,
then squeeze spinach dry.
Brush an 11-by-7-inch rectangular tart pan with a removable bottom (or a 10-inch round pan) lightly with some of the remaining 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 olive oil. Fit into tart pan, leaving a 1-inch overhang. Brush a second sheet with oil and fit into pan. Repeat with remaining sheets, creating a crust that is at least five layers thick over the bottom of the pan. Fold edges at top.
Crumple a double layer of aluminum foil into a rectangle the size of the bottom of the tart and fit into crust to weight 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. (Remove from oven, leaving baking sheet. Tent edges with foil if browning too quick.)
Reduce oven to 375 degrees. In a blender,
puree the goat cheese, eggs, milk, and remaining
1/4 teaspoon salt until smooth. Add
herbs and pulse to combine. Spread vegetables
over crust and pour custard over top.
Place tart back on baking sheet in
oven and bake until custard is set, 20 to 23
minutes. Pull tart from oven and let cool
10 minutes on wire rack. Remove sides
of pan and cool at least 10 minutes more,
until ready to serve. If not serving right
away, slide off of bottom and onto wire rack.