Make or prepare the marinara sauce and keep it covered in the refrigerator until needed.
Whisk the flour, 1/4 teaspoon Kosher salt, and 1/4 teaspoon black pepper in a large bowl.
In another bowl, whisk 4 large eggs, 3/4 cup whole milk, and 3/4 cup heavy cream until combined.
Gradually whisk half of the egg mixture into the flour mixture until smooth, then whisk in the remaining egg mixture until the batter is completely smooth. Chill, covered, at least 30 minutes or up to 12 hours.
Strain the batter through a fine mesh sieve into a bowl, then stir in 2 tablespoons melted unsalted butter.
Heat an 8–9 inch nonstick skillet over medium and lightly coat with canola oil. Pour about 1/4 cup batter and swirl to form a thin crêpe. Cook until set and lightly browned on the bottom, 1–3 minutes, flip and cook about 1 minute more.
Transfer crêpes to a plate with parchment or wax paper between each and allow to cool completely. Repeat until batter is used.
In a large bowl, stir together 1 cup ricotta, 1 cup chopped fresh mozzarella, 1/2 cup shredded smoked mozzarella, and 3 tablespoons of the grated Pecorino Romano. Season with 1/2 teaspoon Kosher salt and 1/4 teaspoon black pepper.
Stir in 2 tablespoons chopped basil and 1 tablespoon chopped parsley, then gently fold in the lightly beaten large egg until smooth. Cover and refrigerate 30 minutes or overnight.
Preheat oven to 375°F (190°C). Spread 1–2 cups marinara across the bottom of a 9 x 13-inch baking dish.
Working one crêpe at a time, spread about 3 tablespoons of the cheese filling onto the lower third of the crêpe and roll to form a stuffed tube. Place each stuffed crêpe seam-side down in the baking dish on top of the sauce.
Spread the softened 2 tablespoons butter over the tops of the crêpes and sprinkle with a couple tablespoons of the remaining grated Pecorino Romano.
Bake on the middle rack 12–15 minutes until cheese is melted and bubbly and edges begin to brown. Optionally broil briefly on the middle rack to crisp edges, watching carefully to avoid burning.
While baking, warm the remaining marinara in a medium pot over medium heat.
Remove from oven and let rest 10 minutes, then top with chopped parsley and serve with warmed marinara and extra grated cheese if desired.