Season chicken breast with salt and pepper and cook for 20-25 minutes in the oven at 350 degrees or grill until completely cooked. When the chicken is cool, shred with two forks and mix with salsa verde. Set aside until peppers are cooked.
While wearing gloves, remove the top of the peppers and cut them down one side. Using a spoon, remove the seeds from the poblanos.
Grill the poblanos on medium heat for about 10 minutes or until the skin starts to char and blister. Turn them over halfway through cooking so they blister evenly.
If you plan to remove the skins, remove the poblanos from the grill and place them in a bowl and cover with plastic wrap for 5 minutes. This will soften the peppers and make the peppers easy to peel.
Open the peppers gently, stuff with the chicken and salsa verde mixture. Top with queso fresco and fold the pepper back over the chicken.
Place the peppers on a sheet pan and place under the stove broiler for 2-3 minutes until cheese is melted and ingredients are heated through.