Melt ½ tbsp butter in a pan over medium heat. After the butter is melted and hot enough, add the chicken thighs and cook for about 3-5 minutes then sprinkle salt and black pepper. After that flip the chicken and continue cooking for about 3 minutes until all the sides of the chicken turn golden brown. Set aside and put it in a bowl.
In the same pan, add the remaining butter. After melted, add the flour and toast until the flour turn a little bit golden.
Add 1/4 part of the milk while mixing so there is no lump on the saucepan. Once blended, add the remaining milk in 1/4 part increments, mixing well after each addition, until all the milk is used and sauce is free from lumps.
After that, add the cooked chicken thighs and stir until mix well. Add salt and blackpepper if necessary. Set aside.
In a cleaned pan, toast tortilla over low heat, sprinkles about 1/5 cup cheddar cheese and about 1/5 cup mozzarella cheese.
Add 1/4 of the bechamel chicken on top of the cheese and sprinkle it with 1/5 cup cheddar cheese and 1/5 cup mozzarella cheese again. Cover it with tortilla.
Toast until the edges of the bottom tortilla change color to light brown. Turn the quesadilla over slowly then toast for another 3 minutes.
After 3 minutes, turn off the heat then transfer the creamy cheesy chicken quesadillas into a cutting board then cut it into four parts.
Toast another 3 quesadillas with the same steps.
Creamy cheesy chicken quesadillas are ready to serve.