In the bowl of a stand mixer, whisk together 1 cup (125 grams) flour, sugar, salt, yeast, and baking soda by hand.
In a small saucepan, heat 3⁄4 cup(180 grams) water and 3 tablespoons (42 grams) butter over medium-low heat until butter is melted and an instant-read thermometer registers 120°F (49°C) to130°F (54°C). Add warm butter mixture and ricotta to flour mixture; using the paddle attachment, beat at medium-low speed until combined, about 1 minute, stopping to scrape sides of bowl. With mixer on low speed, gradually add 2 1⁄4 cups (281 grams) flour, beating just until combined and stopping to scrape sides of bowl.
Switch to the dough hook attachment. Beat at medium-low speed until a soft, somewhat sticky dough forms, 12 to 15 minutes, stopping to scrape sides of bowl and dough hook; add up to remaining 1⁄4 cup (31 grams) flour, 1 tablespoon (8 grams) at a time, if dough is too sticky. (Dough will mostly pull away from sides of bowl.) Turn out dough onto a lightly floured surface, and shape into a ball.
Spray a large bowl with cooking spray. Place dough in bowl, turning to grease top. Cover and let rise in a warm, draft-free place (75°F/24°C) until doubled in size, 30 to 45 minutes.
Punch down dough; cover and let stand for 10 minutes.
In a small microwave-safe bowl, heat remaining 5 tablespoons (71 grams) butter on high in 10-second intervals until melted. In another small bowl, stir together provolone and Parmesan.
Spray a 12-cup muffin pan with baking spray with flour.
On a lightly floured surface, divide dough in half. Roll half of dough into a 12-inch square. (Keep remaining dough covered with plastic wrap.) Brush 1 tablespoon (14 grams) melted butter onto dough. Cut dough into 12 (6×2-inch) strips. Sprinkle 1 tablespoon (8 grams) cheese mixture onto each of 10 strips. Stack 5 strips, cheese side up, on top of each other; place 1 plain strip, butter side down, on top. Repeat with remaining 6 strips of dough. Using a floured serrated knife, cut each dough stack crosswise into 3 (2-inch) squares. Place dough squares, cut side up, in prepared muffin cups. Repeat procedure with remaining dough, 1 tablespoon (14 grams) melted butter, and cheese mixture. Sprinkle any remaining cheese mixture onto dough in cups. Loosely cover dough with plastic wrap, and let rise in a warm, draft-free place (75°F/24°C) until dough fills cups, 15 to 25 minutes.
Preheat oven to 375°F (190°C).
In a small bowl, whisk together egg and remaining 1 tablespoon (15 grams) water; brush onto dough.
Bake until lightly browned,12 to 15 minutes. Let cool in pan on a wire rack for 10 minutes. Stir parsley into remaining melted butter; brush onto rolls. Serve warm.