Preheat griddle to medium. Clean each blossom by removing any sharp green sepals at the base and snap off the stem. Rinse lightly and delicately blot dry.
Place as many corn tortillas on the ungreased griddle as possible without overlapping. Generously sprinkle each with shredded cheese, keeping cheese away from the edge. Scatter 2 squash blossoms, onion slices, some chile, and epazote overthe cheese. Place another corn tortilla on top, press together with a spatula, and cook, turning, until the cheese melts and is stringy and gooey.
Cut each quesadilla into wedges. Serve bubbling hot.