Preheat oven to 375 degrees F.
Melt 3 Tbsp butter in a large cast iron skillet over medium-high heat, then add sage and rosemary and cook until fragrant—2 minutes. Add in ground lamb and begin to brown the meat, stirring minimally to get the best color. Gradually break the lamb into large chunks as you cook it. When lamb is browned, remove from pan and set aside (make sure to keep the juices that render from the lamb).
In the same pan on medium-low heat, add 2 Tbsp more butter and melt. Next, add eggplant, onion, minced garlic, kosher salt, and pepper to taste. Stir occasionally, until onions are translucent and eggplant is very tender, 10-12 minutes. Return lamb back to the pan and gently mix in along with arugula. When ingredients are well incorporated and arugula has wilted, remove from heat and set aside.
In a medium sauce pan, heat butter on medium-low heat and add remaining crushed garlic, rosemary, and sage bundles. Stir and allow herbs to infuse into the oil. After 3 minutes of stirring, add pumpkin puree, Parmesan, heavy cream, and vegetable broth. Simmer ingredients until the sauce thickens, 10 minutes, then whisk in 2 Tbsp flour to thicken. Keep warm and set aside.
Grease a large (12 inch) cast iron skillet with butter. To make the lasagna, you will have three layers of pasta, three layers of mozzarella, three layers of sauce (plus about 1/4 c on the bottom, and two layers of meat)—keep this in mind as you add in each layer.
Layer a small, scant 1/4 cup portion of sauce on the bottom of the pan and cover with a layer of noodles. Spread meat and vegetables on top, followed by more sauce, and a layer of mozzarella. Add one more layer of pasta, meat, sauce, cheese, and the final pasta layer. Spread remaining sauce over the top, followed by mozzarella and remaining 1/4 c Parmesan.
Bake for 30-40 minutes, or until the top has browned. When lasagna is cooked, remove from the oven, and let it rest for 10 minutes before serving with fresh basil. Enjoy!