Preheat oven to 400°F. Line a baking pan with parchment.

Place garlic, rosemary, and salt in a small bowl, and crush garlic into a past with the back of a spoon. Add oil and freshly ground black pepper to taste and stir to combine.

On prepared pan, place a layer of eggplant slices. Brush with oil mixture. Add another layer, and brush with oil mixture. Continue until all eggplant is layered.

Roast in the oven until starting to char, about 30-40 minutes. Let cool.

Reduce oven to 350°F for baking the assembled lasagna.

While the eggplant is roasting, heat oil in a large skillet over medium heat. Add onions, and fry until tender, about 7 minutes. Add garlic and grated carrots and fry until the carrots start sticking to the pan. Add mushrooms and wine and cook until mushrooms are tender and wine is reduce by about half.

Add tomatoes, oregano, bay leaf, and kosher salt. Continue to cook, stirring occasionally, until thick. Add peas and bring to a simmer to heat all the way through. Remove from heat.

In a medium bowl, stir together goat cheese, eggs, basil, and nutmeg. Set aside.

Lightly oil a 9x13 inch pan. Place a light layer of sauce on the bottom, followed by half the eggplant, half the remaining sauce, and half the goat cheese mixture. Layer on remaining eggplant, press smooth, then top with remaining sauce and remaining cheese mixture.

Sprinkle with Parmesan cheese, cover with a piece of parchment, and bake until bubbling through, about 45 minutes. Remove parchment and bake until cheese is browned, 15 minutes more.

Remove from oven and let stand for 15 minutes before slicing and serving.