In a bowl, whisk together the flour, instant yeast, sugar, parmesan, and salt. Add the warm milk, honey, 4 eggs, and yolk. Using a dough hook on a stand mixer, mix together at a low to medium speed until a shaggy dough forms. Mix the dough together for another minute or so until the dough is combined.
Split the butter into four portions. Add the first portion into the dough, and knead for about two minutes until fully incorporated. Scrape down the sides of the bowl, and add in the second portion. Continue this for the next few portions. The batter will resemble cake batter at this point. Increase the mixer speed to medium and knead for an additional 8-10 minutes, or until the dough is completely smooth and velvety. It'll come off the sides of the bowl cleanly when done.
Turn the dough out on to a lightly floured surface. Fold each side into the center and repeat 3 times. Flip the dough over and you'll have a smooth surface. Transfer the dough to a bowl for the first proofing. Cover the bowl and allow it to proof for about an hour, or until it doubles in size.
While the dough rests, toss together all the ingredients for Mom's Green Chutney into the blender, removing the water from the recipe, until a thick chutney forms.
Layer parchment paper in a 10-inch cast iron or spring form pan and butter it. Once the dough has risen, punch it down and turn out the dough onto a lightly floured surface. Roll the dough out into a 12 x 18 inches rectangle. Spread the chutney all over the dough, being sure to go all the way up to the edges. Top with shredded cheese.
Roll the edge of the dough into itself and continue rolling to form a log. Pinch the seals together. Cut the log into 8 rolls. Carefully transfer the rolls into the cast iron. Cover and let rise for another 1-2 hours. Check if the dough is done proofing by poking it. If it leaves an indentation, it's done. Preheat the oven to 450°F.
Brush the top of the rolls with a beaten egg. Bake for 25 minutes until the top begins to brown. Then, reduce the oven temperature to 350°F and bake for another 30-35 minutes until the internal temperature reaches 180°F. Sprinkle additional cheese on warm rolls and serve with additional chutney.