Heat 1 teaspoon oil in a nonstick skillet over medium-high heat. Cook mushrooms for 6–8 minutes until released liquid evaporates and edges brown. Add spinach and stir for 1–2 minutes until wilted. Transfer to a bowl; pour away any watery liquid.

Whisk eggs, milk, salt and pepper. Lower heat to medium-low and add remaining oil. Pour in eggs and move them gently with a spatula until set with no liquid egg, about 3–4 minutes. Egg dishes should reach 160°F (71°C). Transfer to a plate.

Divide cheese, cooked vegetables, scrambled eggs and scallions over one half of each tortilla, leaving a small edge. Fold the empty half over to form a semicircle. Keep the filling layer even rather than piling it in the center.

Wipe the skillet and return to medium heat. Toast two folded quesadillas at a time for about 2–3 minutes per side, until the tortilla is golden and the cheese melts. Lower heat if the tortilla browns before the filling warms.

Transfer to a board and cut each folded tortilla into two or three wedges. Serve one quesadilla per person. Salsa or hot sauce can be offered separately; they are not included in the estimate.