Whisk milk, sugar, butter, oil and 2 eggs in a large bowl. Whisk yeast, whole-wheat flour, 1 1/2 cups cake flour and salt in a medium bowl. Gradually stir the dry ingredients into the wet ingredients using a wooden spoon. The dough will be very sticky.

Sprinkle 1/2 cup cake flour on a work surface. Turn out the dough onto it and knead until all the flour is incorporated, 1 to 2 minutes. Coat a large bowl with cooking spray. Transfer the dough to the bowl, coat the top with cooking spray and cover with plastic wrap. Let rise at room temperature (about 70 degrees F) until doubled in volume, 1 1/4-2 hours.

Coat a 9-by-13-inch metal baking pan with cooking spray. Turn the dough out onto a lightly floured surface; pat into a rough 7-by-10-inch rectangle. Cut lengthwise into 4 equal strips (Photo 1) using a bench knife or butter knife. Then cut each strip crosswise into 6 equal portions (Photo 2). (Each portion will weigh 1 1/4-1 1/2 ounces.)

Working with one portion of dough at a time, gather and pinch the edges together, shaping the dough into a rough ball. The spot where the edges come together is the bottom of the ball (Photo 3). Place each ball, bottom down, on a clean work surface. With a slightly cupped hand, move the ball around in a circular motion, keeping the bottom in place while tucking the loose edges into it and stretching the surface of the dough tight (Photo 4). (If the outer skin breaks, set the roll aside and let it rest while rounding the remaining rolls. Reroll once the dough relaxes.) Arrange the rolls in the prepared pan. Cover with plastic wrap. (If following make-ahead instructions, refrigerate the rolls now.)

Let the rolls rise at room temperature until almost doubled in size, about 1 1/4 hours (Photo 5).

Preheat oven to 400 degrees F. Whisk the remaining egg in a small bowl, and brush the tops of the rolls with it (you'll have some left over); sprinkle with wheat germ. Bake the rolls until light brown on top, about 20 minutes.