Preheat oven to 375°F and line a 11x15 jelly roll pan with parchment paper. Generously brush the parchment paper with the softened butter and set aside. You will also need a clean tea towel to roll the cake up in once it is done baking.
Vanilla Roll Cake
In a medium sized bowl, sift together flour, baking soda, baking powder, and salt. Stir to combine and set aside.
In a large mixing bowl combine the sugar and eggs. Using a hand mixer or whisk, mix on high until the batter is thick and well combined.
Add in the vanilla yogurt, sour cream, vanilla extract and mix again just to combine.
Add in dry ingredients and mix on medium speed for 2 minutes until no clumps remain and batter is very well combined.
Optional step- to make the red stripes on the cake, reserve 1 cup of batter. Add in coloring of choice, stir well to make sure the color is fully distributed, place in a small zip top bag or a disposable piping bag. Do not cut the bag open yet. This batter is very runny and will run out of the piping bag easily. If you are not wanting the stripes skip this step and skip step 8.
Pour the cake batter into the prepared jelly roll pan, and using a slanted spatula, spread it out so it fully covers the entire pan.
Now, if you are adding the stripes, cut a small portion off the end of the piping bag. Be warned that as soon as you cut it open the batter will run out. So seal the end and hold it upwards before cutting off the tip. Staring at one corner, pipe diagonal lines down the batter. (see picture)
Bake the cake for 13 minutes or until it springs back when lightly touched.
Once the cake has baked get your tea towel and ¼ cup of powdered sugar ready.
Immediately after the cake has baked sprinkle ¼ of powdered sugar over the top, cover it with the tea towel, and then cover the tea towel with a large wire cooling rack. Using oven mitts (because the pan will still be hot!) grasp the pan and wire rack tightly and flip the entire thing over.
Now the jelly roll pan will be upside down on top of your wire cooling rack. Remove the jelly roll pan, and carefully peel off and dispose of the parchment paper. Starting at the short end, roll the entire cake up with the tea towel (see picture)
Let the rolled cake cool on a wire cooling rack for at least 45 minutes or until it is cold to the touch. If your cake is too warm it will melt the peppermint buttercream.
Peppermint Buttercream
In the bowl of a stand mixer fitted with the paddle attachment or in a large bowl with a hand mixer, add the cream cheese and butter. Beat on high for 2 minutes and scrape down the sides of the bowl.
1 cup at a time add the confectioners sugar. Beat on high for 2 minutes between each addition, and scrape down the sides of the bowl in between additions.
When all of the sugar is added, scrape down the sides of the bowl and add the milk, vanilla extract, and one or two drops of peppermint extract. Beat on medium-high for 1 minute to incorporate.
Once your cake is completely cooled carefully open it up and unroll it.
Using a spatula spread out all of the peppermint buttercream so the top of the cake is completely covered.
Starting at the same end that you initially rolled the cake up from in the beginning, roll the cake up again. It doesn't have to be super tight, you don't want the buttercream to squeeze out. Note- move slowly for this step, the cake might want to stick to the tea towel, but if you move slowly you can get it rolled up with no rips!
Using the remaining ¼ cup of powdered sugar, coat the outside of the roll and then tightly wrap the roll with plastic wrap. Chill the cake in the refrigerator for a minimum of 4 hours, but overnight is best.
Then remove the plastic wrap and transfer the cake to a cutting board. Using a sharp knife cut the cake into slices and serve. Top with more powdered sugar if desired.