In a medium skillet over medium-high heat, warm 2 teaspoons of the olive oil. Add the green onions and mushrooms and saute for 10 minutes, stirring frequently. Add the thyme and garlic and saute for 3 more minutes. Remove from heat. Stir in 1/4 teaspoon of the salt and the pepper and the cheese. Set aside..
Cut a horizontal slit in each of the chicken breasts, creating a pocket. Do not cut all the way through. Place 1/4 cup of the mushroom/cheese mixture into each of the slit chicken breasts..
Line a baking sheet with parchment paper and set it aside..
Make a "dredging assembly line". Place the flour in a shallow container or on a plate. Next, place the beaten eggs in a shallow bowl. Then place the breadcrumbs on a dinner plate. Set an empty plate after the breadcrumbs..
Carefully working with one chicken breast at a time, holding securely so the stuffing doesn't come out, dredge the stuffed chicken first in the flour, then dip each side in the beaten eggs, and finally set it in the breadcrumbs and pat the crumbs into the egg coating on each side..
Place the coated chicken breast on the clean plate as you repeat the dredging process with the remaining stuffed chicken breasts..
Heat oven to 350°F..
Add the remaining 2 teaspoons of olive oil to the same skillet you used to saute the mushrooms. Heat to a medium-high temperature..
Carefully set the dredged, stuffed chicken breasts in the skillet. (Work with one or two at a time. Don't crowd the skillet.) Allow the chicken to saute for about 5 minutes, carefully flip with a sturdy spatula, and allow the chicken to saute for 5 minutes on the other side..
After chicken is nicely browned on both sides, transfer it to the baking sheet. Repeat browning process with remaining chicken breasts (you may have to add more oil to the skillet), and then bake the chicken for about 20-25 minutes, or until chicken is cooked through..