01 - Melt 1 tablespoon unsalted butter in a medium skillet over medium heat. Add chopped shallot and sliced mushrooms, sautéing for 5 to 7 minutes until mushrooms are golden and moisture has evaporated. Season with salt and pepper. Transfer contents to a plate and wipe skillet clean.
02 - In a mixing bowl, whisk together eggs, whole milk, and a pinch each of salt and pepper until homogeneous. Melt remaining tablespoon of butter in the same skillet over medium-low heat. Add egg mixture and cook gently, stirring frequently, until just softly set, about 2 to 3 minutes. Remove from heat to avoid overcooking.
03 - Lay out flour tortillas on a work surface. Distribute scrambled eggs, sautéed mushrooms, and shredded smoked Gouda over one half of each tortilla. Sprinkle with chopped chives if desired. Fold tortillas over to create half-moon shapes.
04 - Place a large nonstick skillet or griddle over medium heat. Cook quesadillas in batches, 2 to 3 minutes per side, until tortillas are golden and crisp and cheese has fully melted.
05 - Slice each quesadilla into wedges and serve immediately with salsa, sour cream, or avocado slices as desired.