Make the filling:
In a 12-inch skillet, heat 1 Tbs. of the oil and the butter over medium-high heat until sizzling. Add the mushrooms, season with 1/2 tsp. salt and a few grinds of pepper, and cook, stirring frequently, until the mushrooms release much of their liquid and begin to brown, 7 to 9 minutes. Transfer to a medium bowl and return the skillet to the heat.
Add the remaining 2 Tbs. oil, the bell pepper, chile, onion, rosemary, and 1 tsp. salt. Reduce the heat to medium, and cook, stirring occasionally, until the vegetables are very soft and fragrant, 8 to 10 minutes; don’t let them brown.
Sprinkle the flour over the vegetables and cook for a minute or so, stirring and scraping so the flour gets mixed with the fat and starts to toast a bit. Add the broth and let it come to a simmer, stirring and scraping up any browned bits, about 1 minute. Add the tomatoes, lemon juice, and hot sauce; bring to a simmer again and cook for 1 to 2 minutes to slightly thicken the sauce.
Add the chicken and mushrooms and simmer for a few minutes until everything is heated through. Remove from the heat and season to taste with more salt, pepper, hot sauce, or lemon juice. Cover the filling and keep warm.
Make the cheese sauce:
In a heavy-duty 1-quart saucepan, bring the cream to a boil over medium-high heat. Reduce the heat to maintain a lively simmer and cook until the cream has reduced by half, about 10 minutes. Reduce the heat to low and add the cheese, stirring until melted. Season generously with pepper. Remove from the heat and let cool slightly.
Assemble and bake the crêpes:
Position a rack in the center of the oven and heat the oven to 400°F. Butter the bottom and sides of a 9x13-inch baking dish.
Working with a few crêpes at a time, lay them on a clean work surface. Spoon about 3 heaping Tbs. of the filling evenly onto the bottom third of each crêpe. Fold the bottom edge of each crêpe up and over the filling, fold the sides in toward the center, and finish rolling up from the bottom. Evenly arrange the crêpes seam side down in a single layer in the baking dish.
Spread the cheese sauce evenly over the crêpes and sprinkle with the paprika (if using). Bake until the sauce is golden and bubbling slightly, 12 to 16 minutes. Serve.
The filling may also be made up to 3 days ahead. Store it in an airtight container in the refrigerator and reheat over medium-low heat before using it.