Roast the Garlic

1. Place peeled garlic cloves in a small dish. Drizzle generously with olive oil, add a pinch of salt, and cover tightly with foil.
2. Roast in a preheated oven at 140°C (285°F) for 1.5 to 2 hours, until deeply golden and completely soft when pierced.
3. Mash the cooled cloves into a paste. Set aside.

Caramelize the Onions

1. Thinly slice the onions.
2. Cook them slowly in a combination of butter and oil over medium-low heat. Once they begin to develop a golden color, add salt.
3. Continue cooking, stirring frequently, for over 1 hour and 30 minutes, until they are deeply caramelized and sweet. Set aside to cool completely.

Prepare the Dough

1. Mix warm milk with a pinch of sugar and the yeast. Let it activate for 5-10 minutes, until frothy.
2. To the same bowl, add flour, sugar, salt, softened butter, and one egg. Knead the dough for 15 to 20 minutes, by hand or mixer, until smooth and elastic.
3. Place in a greased bowl, cover, and let rise in a warm proofing environment at 28°C (82°F) for 1 hour and 30 minutes, or until doubled in size.

Make the Garlic Butter & Assemble

1. Combine the softened butter with the roasted garlic paste, salt, oil, and chopped parsley. Mix until smooth.
2. Punch down the risen dough and roll it out into a rectangle. Spread the garlic butter evenly over the surface, then sprinkle generously with grated cheese. Top evenly with the cooled caramelized onions. 
3. Cut the filled rectangle into 6 strips, roll each one individually into perfect rolls.
4. Place the rolls in a prepared baking dish, cover, and let them proof again at 28°C (82°F) for 30 to 45 minutes, or until puffy. While they proof, preheat your oven to 180°C (350°F).

 Bake

1. Brush the proofed rolls with a beaten egg wash.
2. Bake for 25 to 30 minutes, until the tops are a deep golden brown and the cheese is melted and bubbly. Let the rolls cool in the pan for about 10 minutes before serving.