Prepare or warm the marinara sauce and set aside; it can be refrigerated for short-term storage.
In a large bowl, whisk together 1 cup flour, 1/4 tsp Kosher salt, and 1/4 tsp black pepper.
In a separate bowl, whisk together 4 large eggs, 3/4 cup whole milk, and 3/4 cup heavy cream until smooth.
Slowly whisk about half of the egg mixture into the flour mixture until smooth, then whisk in the remaining egg mixture until completely smooth. Chill the batter, covered, for at least 30 minutes or up to 12 hours.
Strain the chilled batter through a fine-mesh sieve into a bowl and stir in 2 tablespoons melted unsalted butter.
Heat an 8–9 inch non-stick skillet over medium and lightly oil it. Pour about 1/4 cup batter, tilt and swirl to form a thin crêpe, and cook until set and lightly browned on the bottom, about 1–3 minutes.
Flip the crêpe with a spatula and cook about 1 minute more, then transfer to a plate with parchment or wax paper between each crêpe. Repeat with remaining batter and allow crêpes to cool completely.
In a large bowl, stir together 1 cup ricotta, 1 cup chopped fresh mozzarella, 1/2 cup shredded smoked mozzarella, and about 3 tablespoons of the grated Pecorino Romano; season with 1/2 tsp Kosher salt and 1/4 tsp black pepper.
Stir in 2 tablespoons chopped fresh basil and 1 tablespoon chopped fresh parsley, then gently fold in 1 lightly beaten egg until smooth. Cover and refrigerate the filling for 30 minutes or overnight.
Preheat the oven to 375°F (190°C). Spread 1 to 2 cups marinara sauce across the bottom of a 9x13 inch baking dish.
Working one crêpe at a time, place about 3 tablespoons of filling along the lower third of the crêpe and roll up to form a stuffed tube. Arrange each rolled crêpe seam-side down in the baking dish on top of the marinara.
Spread the 2 tablespoons room-temperature unsalted butter over the tops of the crêpes and sprinkle the remaining grated Pecorino Romano over them.
Bake on the middle rack for 12–15 minutes, until cheese is melted and bubbly and tops are lightly browned. Optionally broil briefly on the middle rack to crisp edges—watch closely to avoid burning.
While baking, warm any remaining marinara in a small saucepan over medium heat.
Remove the baking dish and let the manicotti rest for 10 minutes. Garnish with 2 tablespoons chopped parsley and serve with warmed marinara and extra grated cheese if desired.