Heat up a large skillet over medium heat and add in the olive oil. Add the sweet potatoes allowing them to brown on each side, turning when a nice crust has formed. About 3-4 minutes per side. Remove from the skillet and set aside.
While the sweet potatoes are cooking, cut the cherry tomatoes in half and slice the green onions. Set aside.
In a large mixing bowl, crack the eggs and add the milk. Whisk together until the egg mixture is combined. Season with salt and pepper.
Using the same skillet, melt the butter over medium heat and add in the egg mixture. Scramble the eggs using a spatula and keep stirring as the eggs cook so that they become fluffy. Adjust the seasoning with salt and pepper as needed.
When the eggs are finished cooking, gently stir in the cherry tomatoes and green onions.
Begin to assemble the burritos by adding half the sweet potatoes, half the eggs, half the cheese, and half the green chiles to each burrito down the middle. Fold in the shorter sides and then wrap the burrito lengthwise.
Place the burritos seam side down on the same hot skillet, allowing each side to get golden brown. About 1-2 minutes per side. Remove from heat and cut each burrito in half. Sprinkle some cilantro and serve with a side of salsa.