Bring the water to a boil in a medium saucepan over medium high heat.

Stir in the rolled oats, reduce the heat to low, and simmer for 5 minutes, stirring occasionally, until the mixture begins to thicken.

Add the grated carrots to the saucepan and cook for 5 more minutes, stirring, until the carrots soften and the mixture is creamy.

While the oats and carrots cook, whisk together the apple sauce, honey if using, and egg in a medium bowl until smooth and slightly foamy.

Slowly spoon the hot oat and carrot mixture into the apple sauce mixture, whisking constantly, until fully combined, then stir in the pinch of salt.

Pour the combined mixture back into the saucepan and cook over low heat for 5 minutes, stirring often, until thick and spoonable and the egg is fully set.

Remove from the heat and let the mixture cool for 5 minutes until warm, not hot, then portion into bowls and serve to your dog.