Preheat the oven to 400 degrees F.
In a large pot, bring 3 quarts salted water to a boil. Place the fennel wedges in the boiling water and cook until fork tender, about 6 minutes.
Drain the fennel and transfer to a large baking sheet lined with foil. Brush the fennel with 2 tablespoons olive oil, salt, pepper and 2 tablespoons Parmesan. Bake until lightly browned, about 30 minutes, tossing occasionally.
Meanwhile, bring 3 quarts water to boil in a large pot and about 3 inches water to boil in a medium saucepan. Add the pasta to the large pot and cook according to package directions. Add the vinegar to the saucepan and reduce the heat to low, bringing the water to a gentle simmer.
Crack the eggs into two small ramekins and gently place each egg in the simmering water so that they are not touching. Simmer until the egg whites are set, but the yolk is still runny, about 5 minutes. Using a slotted spoon, gently remove the eggs from the water and place on a paper-towel lined plate to drain.
Drain the cooked fettuccine and toss with the caramelized fennel, the remaining 1 tablespoon olive oil, salt and pepper.
To serve, divide the pasta between two plates, topping each with half the remaining Parmesan, Manchego cheese, a poached egg, and salt and pepper to taste.
A viewer or guest of the show, who may not be a professional cook, provided this recipe. It has not been tested for home use.