Preheat the oven to 375°F (190°C).
Trim the ends from each zucchini and cut each into two 3–4-inch sections. Using a vegetable slicer, mandoline, or the thin blade attachment on a stand mixer, slice each section into one long thin sheet about 8–9 inches long; discard the ends. You should have about 12 sheets total.
Arrange the zucchini sheets in a single layer on paper-towel–lined baking sheets with minimal overlap. Sprinkle about 1/2 teaspoon kosher salt over the tops and let sit 10 minutes until beads of moisture form; blot dry with paper towels. Flip the sheets, salt the other side, wait briefly, and blot dry again to remove excess moisture.
In a mixing bowl, combine the ricotta, mascarpone, and Parmesan with the egg; stir until smooth. Fold in the chopped spinach and minced garlic. Season with 1/4 teaspoon kosher salt and 1/4 teaspoon black pepper, tasting carefully because the zucchini and cheeses add saltiness.
Spoon about 3/4 cup marinara sauce into the bottom of a 13x9-inch baking dish and spread into an even layer.
Place one zucchini sheet on a work surface and spread 2 to 3 tablespoons of the cheese-spinach filling evenly over the sheet. Roll the zucchini starting from the short end toward you and place seam-side down in the prepared dish. Repeat with remaining sheets, arranging the rolls in two rows of six with a little space between them.
Spoon the remaining marinara sauce over the rolls and sprinkle the grated mozzarella evenly on top.
Bake for 20 to 25 minutes, until the sauce is bubbling and the cheese is melted. Let rest 5 to 10 minutes before serving warm.