In a food processor, combine the candy canes and pulse them until they're a fine dust. Place the candy canes into a bowl, and stir in one tablespoon of oil.
Using a stand mixer, cream together the butter and a 1/4 of sugar. Add in the candy cane mixture. Then cream together for one minute.
Add in the vanilla extract and peppermint extract, and whisk for 30 seconds. In a small sauce pan, on medium heat, combine the two eggs and 3/4 cup of sugar. Whisking constantly, cook the eggs until they become frothy, and reach 160° Fahrenheit.
Remove the eggs from the heat, and let them cool about 5 minutes.
Then turn the stand mixer onto low, and slowly pour in the eggs.
Turn the speed up to high, and mix for one minute.
Then turn the speed down to low, and slowly add in the heavy cream. Gently increase the speed, as close to full speed, without the cream splattering everywhere.
Whisk the heavy cream on medium-high for ten minutes, to help the cream thicken.
Then place the cream mixture directly into the freezer for 30 minutes, to help it thicken further.
Once 30 minutes is up, remove the cream mixture from the freezer and mix again for another 3 minutes.
Pour the cream into a large loaf tin, and place in the freezer for at least two hours, or until set.