The developers of Baldur’s Gate 3 designed most questlines so that they may be completed in multiple ways. Thus, giving the players more freedom in how they approach situations. This guide represents one method that is confirmed to unlock this achievement.
You will first encounter the Owlbear cub in the Owlbear Nest in Act 1. The next is just east of Blighted Village near the river. If you kill the Owlbear, don’t kill the cub.
Find Scratch just outside the Blighted Village in Act 1. Interact with him and convince him to meet you at your camp.
When you reach the Goblin Camp, the goblins will be playing a chicken game with the cub. Participate in the game and help it complete the course. Next, take out all 3 goblin leaders (don’t kill Sazza) and clear the goblin camp.
After you wipe out the goblins and take multiple long rests, the cub will approach you in your camp. Offer it food, but be ready for Gale to accidently scare it away.
The next time you take a long rest, the Owlbear cub will return with a wounded paw. Take care of it and the cub will decide to stay at camp. The next long rest, you will wake up to find Scratch and the Owlbear cub next to each other. Pet Scratch, then pet the cub for the You Have Two Hands for a Reason achievement to pop.