Preheat the oven to 400 degrees F and prepare a baking sheet with parchment.
In a large bowl, whisk together the flour, sugar, baking powder, and salt.
Add the cold cubed butter to the flour, toss to coat. Using a pastry cutter, cut the butter and flour until they are pea size. You can also use two knives to cut the butter and flour.
Add the mini chocolate chips to the mixture and mix until combined.
In a small bowl, stir the milk and egg yolks just to blend. Add this mixture all at once to the flour mixture. Gently stir with a fork to begin combining the wet and dry ingredients then add in the candy cane pieces and gently mix, just enough to incorporate.
With your hands to gently knead the mixture together until all the dry ingredients are absorbed into the dough and it can be gathered into a moist, shaggy ball. Do not over knead.
On a floured surface, pat the dough into an 8" circle and using a 3" round cutter, cut dough into rounds. Place each round on a parchment-lined baking sheet. Brush the dough with the egg whites mixed with 1 tsp. of water.
Alternately, turn the dough onto the parchment-lined baking sheet and pat down to make the dough 1" thick and about 6-8" diameter. You want a thick scone.
With a sharp knife cut the dough into 8 wedges and slightly separate. Brush the dough with a beaten egg white mixed with 1 tsp. of water.
Refrigerate the dough for 15 minutes before baking.
Bake for 18 - 22 minutes until the scones are nicely browned.