Preheat grill or a grill pan to medium.
Trim ends of zucchini and slice lengthwise into 1/4 inch thick slices. You should get about 16 long slices of zucchini.
Set zucchini on grill and grill until tender and much of the moisture has been removed, 3 to 5 minutes per side. Remove from grill and set aside on layers of paper towel.
Preheat oven to 350F and lightly grease a 9x12 inch glass or ceramic baking dish.
Add oil to a large skillet set over medium heat. Add garlic and sauté until fragrant, 30 seconds. Add chopped spinach and cook until wilted but still bright green.
In a large bowl, combine ricotta, Parmesan, egg, salt, and pepper. Add spinach mixture and stir until well combined.
Take a zucchini slice and spread a heaping tablespoon of the ricotta filling over 1/2 of it. Roll up tightly and place into the prepared baking dish. Repeat with remaining zucchini and filling.
Spread pesto over zucchini rolls. Sprinkle with mozzarella and bake 30 minutes, until cheese is melted and bubbling.