Preheat oven to 375. Pour tomato sauce into the bottom of a medium baking dish or pie plate.
Brush one side of each slab of eggplant with a light layer of olive oil and lightly salt. Grill in batches (either in a hot grill pan or outdoor grill) for 30-60 seconds per side or until well-marked. If you’ve got a bacon press (it’s a pig!), panini press or small, heavy skillet to use as a weight, now is the time to use it.
Repeat with zucchini, grilling 1-2 minutes/side before turning. Set veggies aside.
Meanwhile, combine all filling ingredients in a medium bowl. Combine all topping ingredients except olive oil in a second bowl.
Lay one piece of zucchini on a cutting board or work surface. Top with one piece of eggplant. Add heaping spoonful of filling to one end of eggplant and roll toward other end to make an involtini. (Make sure the outer zucchini layer gets rolled up, too.) Arrange in baking dish on tomato sauce seam-side down. Repeat with remaining eggplant/zucchini/filling.
Sprinkle panko topping mixture generously over involtini. Drizzle liberally with olive oil. Bake, uncovered, for 25-30 minutes or until topping is browned and sauce is bubbly. Let rest for 5 minutes before serving with additional parsley for garnish.