Head from Astrid down into the Sanctuary. To learn more about who you are going after, you need to talk with Festus Krex. He will explain that he has a lead on the mysterious Gourmet. He came across a signed copy of the Gourmet's cook book. It was made out to Anton Virane. Krux wants you to find Virane, learn the identity of the Gourmet, kill Virane (loose ends and all), then find and kill the Gourmet. After that, you are to obtain the Writ of Passage that the Gourmet has.
To start this off, you need to head for Markarth and make your way to Understone Keep. Once there you will need to make your way to the kitchen. There you will find Anton Virane with his two assistants: Rondach and Voada. Approach and talk with Virane.
Anton will be himself as you start talking with him. He always start with telling people that he is from High Rock and not Skyrim. Go and talk with him. Press him just about about the identity of the Gourmet and he starts to fold. Go for the intimidation and he will just give in and tell you who the Gourmet is. He will explain that the Gourmet is an Orc named Balagog gro-Nolob who presently resides in the Nightgate Inn. Either agree to let him go or stay silent. Go hide in one of the corners of the kitchen after that. Wait until after 1 am. Anton will go to bed. Just wait for the others in the kitchen to leave the area he is. Now kill him and slip back outside. Time to head for the Nightgate Inn.
Head for this out of the way Inn to the West of Windhelm. Where Balagog gro-Nolob is dependent on the time of day. During the night, he will be found down in the basement of the Nightgate Inn. During the day, he will be wandering around just outside the Inn.
During the day, Balagog gro-Nolob can be found around noon at the nearby pond. There he will be fishing for a while. IF it is day time and close to Noon, this is a favorable place to kill him. Just find him while he is fishing. Talk to him to reveal your knowledge or just kill him. Once you have murdered him, grab the Writ of Passage, to fulfill the full contract, just drag his body into the pond. You can also use Unrelenting Force to push the body in as well.
At night you will need to find Balagog gro-Nolob inside the Inn. Get inside the Inn and then make your way to the back of it. You will find a set of stairs leading down to the basement. Once inside the basement, you just need to find your target: Balagog gro-Nolob. Here, it helps to talk with him because he will run and hide behind a haystack. All you need to do is kill him there. His body will be hidden and then all you need to do is take the Writ of Passage from him.
With the Writ of Passage in hand, you are ready for the next part of your assignment. Make your way back to the Dark Brotherhood Sanctuary. Report back to Festus Krex when you return and to show him the Gourmet's Writ of Passage. If you hid the body of the Gourmet, then you will also receive the Nightweaver's Band.