Combine the flour, yeast, sugar, garlic, and salt in a large mixing bowl or the bowl of a stand mixer.
Heat the butter, milk, and water in the microwave or on the stovetop until very warm (120° to 130°F/49° to 54°C). Add this mixture, the egg, and the cheese to the flour mixture. Beat the mixture with a handheld mixer or a stand mixer on medium speed. Cover and let rest for 10 minutes.
To make the topping:
Using an electric mixer on medium speed, combine the cream cheese, Parmesan cheese, milk, rosemary, thyme, and garlic until thoroughly combined. Set aside.
To assemble and finish the rolls:
Grease a 12-cup standard muffin pan.
Divide the batter evenly among the muffin cups (about 2 to 3 tablespoons of batter in each). Top each with a rounded teaspoonful of the topping mixture, and spread it gently over the top of each roll. Cover the rolls and let them rise in a warm, draft-free area until they have doubled in size (45 to 60 minutes).
Preheat oven to 375°F (190°C).
Bake the rolls 15 to 20 minutes, or until golden brown. Cool the rolls on a wire rack.