Position oven rack in lower third of oven. Heat oven to 400°F.
Lift dough into 10-inch tart pan with removable bottom; ease dough into pan and up sides; do not stretch; trim edges. Prick bottom all over with fork. Line with foil, gently pushing foil against side of pan. Fill foil with dried beans or pie weights.
Bake crust in 400°F oven for 15 minutes. Remove from oven. Carefully remove foil and beans. Cool on wire rack. Leave oven temperature at 400°F.
Heat oil in large nonstick skillet over medium heat. Add onion, and cook until softened slightly, for about 10 minutes. Add mushrooms, thyme, salt and pepper; cook over high heat until the vegetables are tender and any liquid has evaporated, for about 3 to 4 minutes.
Combine eggs, milk and nutmeg in bowl. Add mushroom mixture.
Brush bottom of pastry shell with mustard. Sprinkle with cheese. Scrape mushroom mixture into shell.
Bake in lower third of 400°F oven 20 to 25 minutes, until center is just set, top is golden. Transfer to rack to cool slightly. Garnish with thyme.