To prepare pastry, lightly spoon flours into dry measuring cups; level with a knife. Combine flours and 1/2 teaspoon salt in a medium bowl. Cut in butter with a pastry blender or 2 knives until mixture resembles coarse meal. Combine sour cream and juice in a bowl, stirring well. Stir sour cream mixture into the flour mixture. Sprinkle surface with 3 tablespoons ice water, 1 tablespoon at a time; toss with a fork until moist and crumbly (do not form a ball).
Gently press mixture into a 4-inch circle on plastic wrap; cover. Chill 20 minutes. Slightly overlap 2 sheets of plastic wrap on a slightly damp surface. Unwrap and place chilled dough on plastic wrap. Cover dough with 2 additional sheets of overlapping plastic wrap. Roll the dough, still covered, into a 12-inch circle (about 1/4-inch thick). Place dough in freezer 5 minutes or until plastic wrap can be easily removed. Remove top sheets of plastic wrap; place dough, plastic wrap side up, onto a baking sheet lined with parchment paper. Remove remaining plastic wrap.
Preheat oven to 400°.
To prepare filling, thinly slice white parts of onions, reserving green tops. Heat a large nonstick skillet over medium-high heat. Coat pan with cooking spray. Add sliced onions to pan; sauté 15 minutes or until golden. Add mushrooms to pan; cook 5 minutes or until moisture evaporates. Remove from heat. Slice green tops of onions. Add sliced tops, 1/4 teaspoon salt, and pepper to pan; stir to combine. Spread onion mixture evenly over dough, leaving a 2-inch border; sprinkle cheese evenly over onion mixture. Fold edges of dough toward center, pressing gently to seal (dough will only partially cover onion mixture). Place pan on bottom rack in oven; bake at 400° for 25 minutes or until browned.