In a large skillet, heat oil over medium-heat. Add mushrooms and sauté for 5 minutes, stirring occasionally.
Add kale and sauté for another 5 minutes, stirring occasionally. (Sometimes, I add a few teaspoons of water while cooking kale to help steam it.)
While kale and mushrooms are cooking, in a small bowl, whisk together eggs, cheese, milk and black pepper.
Turn heat up to medium-high and pour egg mixture into skillet with mushrooms and kale. Stir eggs with spatula frequently to scramble and mix everything together. Continue to scramble eggs until no longer runny, about 2 – 3 minutes.
Remove from pan and divide egg mixture into two tortillas. Wrap and serve.