Preheat oven to 350 degrees F. Grease a 18 x 13 inch rimmed baking sheet or line with parchment paper.
In a large bowl, combine the flour, sugar, baking soda, and salt. In another bowl, whisk together the eggs and yolks, sour cream, vanilla extract, and peppermint extract until smooth.
In a large saucepan over medium heat, heat the chocolate, butter, oil, water, and cocoa, stirring occasionally, until smooth, 3 to 5 minutes. Whisk the chocolate mixture into the flour mixture until incorporated. Whisk the egg mixture into the batter until well-combined, then pour into the prepared baking pan.
Bake 18 to 20 minutes or until a toothpick inserted into the center comes out clean. Transfer the pan to wire rack.
ICING
About 5 minutes before the cake is done baking, heat the butter, cream, cocoa, and corn syrup in a large saucepan over medium heat, stirring occasionally, until smooth. Pull the pan off the heat, and whisk in the powdered sugar, vanilla and peppermint extract.
Spread the warm icing evenly over the hot cake. Let the cake cool to room temperature on the wire rack (about one hour) then sprinkle with chopped candy canes.
Refrigerate until the icing is set (about 1 hour longer) before cutting into squares. The cake can be wrapped in plastic and refrigerated for up to 2 days. Bring to room temperature before serving.
Notes
The key to perfectly moist Texas sheet cake is to let the warm icing soak into the hot cake. As soon as the cake comes out of the oven, pour the warm icing over the cake and use a spatula to spread the icing to the edges of the cake. This creates the signature fudgy layer between the icing and the cake.