Carefully peel 6 outer leaves off each head of cabbage and set aside (you'll have 12 leaves total). If you are having difficulty peeling the leaves, trim off part the root end and peel from the bottom. Cut the remaining cabbage into quarters then core and shred.
Bring a large pot of water to a boil over high heat. While you are waiting for the water to boil, prepare an ice bath (fill a large bowl with ice and cold water). Once the water is boiling, add a few large pinches of salt. Add 6 of the outer cabbage leaves to the boiling water and cook for 2 minutes, then transfer to the ice bath. Repeat with the remaining 6 outer cabbage leaves. Once cooled, transfer the cabbage leaves to a paper towel-lined baking sheet (you will probably need 2).
Preheat an oven to 350°F. Butter a 9 x 13 baking dish and set aside.
In a large, heavy bottomed pot, warm extra virgin olive oil over medium heat. Add the bacon strips and cook, stirring occasionally, until golden and crispy. Add the garlic and cook for 1 minute. Add the shredded cabbage, tossing to coat in the oil (I used my tongs here). Cook the cabbage until tender and wilted, stirring occasionally (7 - 8 minutes). Season with kosher salt and freshly ground black pepper then let cool slightly.
To a large bowl add eggs, bread crumbs, 1/4 cup grated Parmigiano-Reggiano and the cabbage mixture, stirring to combine.
Lay the outer cabbage leaves out on a work surface with the inside facing up. Spoon the filling evenly on to the center of each leaf. Roll up each leaf similar to rolling a burrito (fold up the bottom up, then fold in the sides, then roll into a tight cylinder). Place each role, seam side down, into the prepared baking dish. Cover with foil then bake for 30 minutes.
After 20 minutes, start preparing the sauce. In a small skillet, warm butter over medium low heat. Cook until the butter starts to brown (5 - 7 minutes). You will know the butter is done when it gives off a nutty aroma and you see small brown flecks swimming around. Take the skillet off of the heat and add the rosemary leaves.
Take the baking dish from the oven and uncover. Sprinkle the stuffed cabbage with the remaining 3/4 cup cheese then drizzle with rosemary brown butter.