Use a mandoline and slice zucchini longwise about 1/8 inch thick.
Lay the zucchini out on a baking sheet and sprinkle with 1 tablespoon salt. Let it set for about 15 minutes, rinse and pat dry with paper towels.
Spread the zucchini onto a dry baking sheet and bake at 300° for 10 minutes. This will help keep the lasagna from being watery. It is optional and for this recipe, we skipped the salting and baking the zucchini. The zucchini wasn't mushy at all but it was not as firm as traditional lasagna.
In a bowl, mix ricotta with 2 cups of mozzarella, egg, 2 teaspoons oregano, 2 teaspoons basil and 1 teaspoon garlic.
Use a medium sized bowl to make the sauce. Combine the diced tomatoes with the tomato sauce, 1 tablespoon oregano, 1 tablespoon basil, 1 teaspoon garlic and 1 teaspoon salt.
Add the brown sugar (skip if making low-carb) and grated carrot, mixing well.
Spoon enough sauce into a 9 x 9 inch baking pan to cover it.
Layer with zucchini, filling the pan, it is fine to overlap.
Spread with filling in an even layer.
Add another layer of zucchini, sauce and ricotta.
Repeat the layers, once more, ending with the sliced zucchini.
Top with mozzarella and grated parmesan.
Place on preheated grill at 300° and cook with the grill closed 1 1/2 hours.
Remove from grill and allow to rest 15 minutes before slicing and serving.