Lay out 2 of the flour tortillas on a flat surface. Top with the cheeses, spreading evenly over the tortillas. Top with remaining tortillas and press to seal.
Heat a large saute pan over high heat. Add enough oil to coat the bottom of the pan and lower the heat to medium. Cook until golden brown on both sides, about 3 minutes per side. Repeat with the other quesadilla.
Let cool for 5 minutes. Slice into 8 or 10 pieces. Serve with the guacamole.
Halve the avocado and remove the pit with a sharp knife. Discard the pit. Scoop the flesh into a small bowl and mash with a fork. Stir in the remaining ingredients, season with salt to taste, and set aside.