To prepare crust: Whisk whole-wheat flour and all-purpose flour with salt in a medium bowl. Cut butter into small pieces; using your fingers, quickly rub the butter into the dry ingredients until smaller but still visible.

Add sour cream and oil; toss with a fork to combine with the dry ingredients. Sprinkle 2 tablespoons of ice water over the mixture. Toss with a fork until evenly moist; if the mixture seems dry, add up to 1 more tablespoon water. Knead the dough in the bowl a few times--the mixture may still be a little crumbly--then firmly press into a disk. Cover the bowl with plastic wrap and refrigerate for at least 1 hour.

To prepare filling & bake quiche: Preheat oven to 375 degrees F. Coat a 9-inch pie pan with cooking spray.

Heat oil in a medium skillet over high heat. Add onions and 1/8 teaspoon salt; cook, stirring frequently, until the onions start to brown, 3 to 5 minutes. Add water, reduce heat to low and cook, stirring frequently, until the onions are golden brown and very soft, about 15 minutes. Remove from heat and let cool while you roll out the crust.

Place the dough on a sheet of parchment or wax paper and roll into a 12- to 13-inch circle, dusting the top with a little flour, as needed. (If chilled more than 1 hour, let the dough stand at room temperature for 5 minutes before rolling.) Place the prepared pie pan upside down in the center of the dough. Holding one hand on top of the pan and the other hand underneath the paper, flip pan and dough over so the dough is lining the pan. Remove the paper and patch any tears in the dough. Trim the crust so it evenly overhangs the edge by about 1 inch, then tuck the edges under at the rim and crimp with your fingers or a fork.

Spread the caramelized onions in the bottom of the crust. Layer bell pepper and ham over the onions and sprinkle with thyme. Top with cheese. Whisk eggs, egg whites, milk, sour cream, pepper and the remaining 1/4 teaspoon salt in a medium bowl. Pour the mixture into the crust.

Bake the quiche until puffed and firm when touched in the center, 40 to 50 minutes. Let cool on a wire rack for 15 minutes. To serve, cut into 8 pieces.