Add spinach and chard, stir, and cook until wilted, 1–2 minutes. Remove pan from the heat and season with more salt and pepper, to taste. Set aside until crepes are ready.

Bring eggs and 1 cup of milk to room temperature, then combine with spinach, sugar, and salt in a blender. Once the mixture is smooth, pour it into a large bowl, sift in flour, add baking soda, and whisk together. In a small saucepan, bring remaining cup of milk to a boil. Add oil to batter and whisk, then add boiling milk and whisk again until bubbly.

Heat a crepe pan (or nonstick or cast-iron skillet) over medium heat for 3 minutes and turn stove down to medium-low. Spray lightly with oil and pour about 1/2 cup of batter onto the pan. Using pot holders as needed, pick up the pan and tilt so the batter spreads out to form a 10-to-12-inch circle. Cook undisturbed for about 90 seconds, until crepe is covered with tiny bubbles, the edges begin to pull up from the pan, and the underside is golden. Using a large, flat spatula or crepe turner, flip the crepe, then cook for 30 seconds (any longer and you will lose the vibrant green color). Place crepe on a plate, rub butter over the top, and make the next crepe, turning the stove down as needed. As you go, continue to stack the crepes, green side up (this will keep the crepes from sticking together). Cover the stack with a kitchen towel until all crepes are cooked.

Heat skillet over medium heat and place a crepe golden side up. Fill with the mushroom-spinach mixture (about 1/3 cup) and sprinkle some cheese on top. Fold crepe in half, then into quarters, and remove from pan. Rapidly prepare the rest of the crepes and serve immediately. If needed, finished crepes can be held, covered, in a warm oven (175°F), while preparing the remaining crepes.