While Fuzzy Search is a short Lightning Returns: Final Fantasy XIII side-quest and with a small payout, you should focus on completing it as soon as possible, especially if you want to heal Lightning’s Chocobo faster.
For finishing the said quest, along with other rewards, you will get the Fuzzy Sheep Milk, required for the Saving An Angel side-quest.
Fuzzy Search can be accessed between 6 AM and 6 PM, and you don’t have to complete other quests in order to activate it.
Because of this, you should address it while working on finishing Saving An Angel, and the following walkthrough explains how to complete the quest’s objectives.
Description: “Disaster has struck in the village of Aryas – three of Dr. Sheep’s prize specimens have broken out of their pen! Can Lightning round up the strays?”
Location: The Wildlands
Requirements: Day 1. Start the quest at 6AM, and finish it before 6PM
Rewards: 400 Gil, Maximum HP + 40, Strength + 6, Fuzzy Sheep Figurine Adornment, Canned Sheep Logo Adornment
For Fuzzy Search you have to travel to the Aryas Village, after you get the food for your Chocobo, from the citizens in Canopus Farms. Aryas Village can be found south from Canopus Farms, but since it is a long trip, it is a good idea to keep the Chronostasis activated.
Open your map and SE from Canopus Farms you will see a pass towards Aryas Climb. Cross the pass then head south from Aryas Climb and make sure you remain in the central area.
Soon you will see the Aryas Village, where you must find Dr. Sheep.
Look around for a corral and in the middle you should see a NPC wearing glasses and a violet jacket. Talk to him and start the quest. Your goal is to bring three fuzzy sheep back to the corral. Notice that on your minimap they are marked by yellow dots.
The three specimens are: Shaken Sheep, Fearful Sheep and Scared Sheep. Press RB on Xbox 360 to draw your weapon, and guide each sheep back to the corral, by attacking them.
When the last sheep enters the corral, talk to Dr. Sheep to finish the side-quest and get all rewards along with the Fuzzy Sheep Milk.
Make sure that you give the Fuzzy Sheep Milk to your injured white Chocobo waiting for you at Canopus Farms.