Place potatoes in a medium pot and cover with water. Bring to a boil over high heat, then reduce to medium and let potatoes simmer until very tender, about 15 minutes. Drain potatoes and let cool 5 minutes then finely mash with roasted garlic until no lumps remain. Set aside to cool.
In the bowl of a large stand mixer fitted with a dough hook attachment, dissolve yeast in warm water. Let stand 5 minutes until foamy, then mix in honey, oil, egg, salt, and cooled potato mixture.
Add 4 cups flour to bowl and mix on medium speed until a smooth dough forms, adding additional flour if needed. Turn dough onto a floured surface and knead until smooth and elastic, about 5 minutes. Place dough in a large greased bowl, turning once to grease top. Cover bowl with plastic wrap and let rise in a warm place 1-1/2 hours or until doubled in size.
Punch risen dough down and divide into 16 pieces. Roll each portion of dough into a smooth ball and place in a greased 9x13 baking pan.
In a small bowl, whisk egg and water together until smooth and brush over dinner rolls dough. Let rolls rise until doubled in size, about 30 minutes.
Bake dinner rolls at 375F 20-25 minutes or until rolls are a deep golden brown. Remove from oven and sprinkle with Parmesan cheese. Return to oven and broil on high 1-2 minutes until cheese is melted. Remove rolls from oven and allow to cool 10 minutes then serve warm. Enjoy!