Put the poblano under the broiler for about 10 minutes, turning once until it blackens. Place in a plastic bag, close it and let it sit for about 20 minutes. After this time has passed, take poblano out of the bag, peel it (skin should shred off easily), remove stem and seeds and dice.
Gently wash the squash blossoms (there might be bugs inside) and remove stems and stamens. Roughly chop.
Heat skillet to medium and add the olive oil. Add onions and diced poblano and cook for about 5 minutes or until onions are translucent.
In a skillet heated to medium, melt a tablespoon of butter. Add a tortilla and cook it on one side until it puffs (about 30 seconds). Flip tortilla over and sprinkle over entire surface 1/4 cup of squash blossom filling and 1/2 cup of grated cheese.
Top with another tortilla, and after cheese has melted and the 2 tortillas stick together (a couple of minutes), flip quesadilla and cook for a couple of minutes more or until lightly browned.
Repeat for the remainder of the filling and tortillas. Serve warm with salsa verde on the side, if you like.