How do I feed my sylesti a Magical Pie? I'm reading the description of the pie, (in my inventory) but it says to press "Eat me" on the profile page. So I looked on my sylesti's profile page, but there was no "Eat me" button.

You can find magical pies under "philters". So click that, find it from the drop down list, and click use.

Note that your pet's age on the profile will remain the same, since that tracks the amount of days since the pet was created. What increases is the maturity, which determines what stage the pet is at. You can only see a pet's maturity if it has been genetically tested, and by looking at the image code. So if your pet is not tested, you won't be able to see the actual maturity, but the pie will increase it, causing your pet to reach the next stage faster.