1/2 teaspoon dry yeast
2 tablespoons warm water (100° to 110°)
2 tablespoons extra-virgin olive oil, divided
1 teaspoon dried thyme
1/3 cup 2% reduced-fat milk
1/2 cup (2 ounces) grated Parmigiano-Reggiano cheese, divided
1 tablespoon sugar
1/2 teaspoon kosher salt
1 large egg, lightly beaten
1 1/10 ounces whole-wheat white flour (about 1/4 cup)
5 3/5 ounces all-purpose flour (about 1 1/4 cups), divided
Cooking spray
1/2 teaspoon cracked black pepper

Dissolve yeast in 2 tablespoons warm water in a large bowl; let stand 5 minutes or until bubbly.
Heat 1 tablespoon oil in a small saucepan over medium heat. Add thyme to pan; cook 1 minute or until bubbly and fragrant. Add thyme mixture and milk to yeast mixture, stirring with a whisk; add 1/4 cup cheese, sugar, salt, and egg, stirring well.
Weigh or lightly spoon whole-wheat white flour into a dry measuring cup; level with a knife. Using a wooden spoon, stir whole-wheat white flour into yeast mixture. Weigh or lightly spoon 4.5 ounces (about 1 cup) all-purpose flour into a dry measuring cup; level with a knife. Add all-purpose flour to yeast mixture, stirring well. Add enough of remaining all-purpose flour, 1 tablespoon at a time, to form a smooth but very sticky dough. Place dough in a large bowl coated with cooking spray, turning to coat top. Cover and refrigerate overnight. (Dough will not double in size.)
Remove dough from refrigerator. Do not punch dough down. Turn dough out onto a floured surface; sprinkle dough lightly with flour. Roll dough into a 12 x 7–inch rectangle. Brush dough with remaining 1 tablespoon oil. Sprinkle remaining 1/4 cup cheese evenly over dough; sprinkle with pepper. Beginning with a long side, roll up dough jelly-roll fashion. Pinch seam to seal (do not seal ends of roll). Cut roll into 8 (1 1/2-inch) slices. Place slices, cut sides up, on a baking sheet covered with parchment paper. Cover and let rise in a warm place (85°), free from drafts, 1 hour or until rolls have risen slightly.
Preheat oven to 400°F.
Place pan in oven, and immediately reduce heat to 375°. Bake rolls at 375° for 12 minutes or until golden brown. Serve warm.