Make the aioli: Heat the paprika and vegetable oil in a small saucepan over low heat. When hot, remove the pan from the heat and allow the oil to steep at room temperature, 20 minutes. Transfer to a small bowl and refrigerate at least 1 hour.
Combine the egg yolk, vinegar and lemon juice in a food processor or blender; process until smooth and frothy. With the machine running, slowly pour in the paprika-infused oil and process until the desired consistency is reached. Season with salt. (The aioli can be made up to 3 days ahead; refrigerate in an airtight container.)
Make the filling for the peppers: Combine the mayonnaise, creme fraiche, mustard, lemon juice, manchego, parsley, chives and egg in a large bowl; mix thoroughly with a rubber spatula to combine. Gently mix in the crabmeat (do not break it up too much). Season with salt and pepper.
Drain the piquillo peppers thoroughly. Remove the seeds, making sure to keep the peppers whole and intact. Carefully spoon about 1 tablespoon crab filling into each pepper. Transfer to an ovenproof baking dish, such as a cazuela. (The peppers can be filled 1 day ahead; cover and refrigerate.)
Preheat the oven to 375 degrees F. Transfer the peppers to the oven and bake, uncovered, until heated through, about 15 minutes. Top the peppers with the aioli and return to the oven until the aioli is hot, about 2 more minutes.