In a sauté pan over medium heat, warm 1 Tbsp. of the olive oil.
Add the shallot and cook, stirring occasionally, until softened, 2 - 3 minutes. Add the garlic and
cook for 1 minute. Add the spinach and cook, stirring occasionally, until softened, 1 - 2 minutes. Season with salt and pepper.
Remove from heat and let cool until barely warm. Stir in the cheese.
Preheat an oven to 400°F/200°C.
Season the chicken on both sides with salt and pepper. Create a pocket in each chicken breast: lay the breast on a work surface with the pointed end away from you. Insert a long paring knife in the center of the end closest to you (the rounded end). Cut around the thicker side of the breast, stopping 1/2 inch short of the opposite side, being careful not to cut through to the outside of the breast.
Repeat with the remaining chicken breasts. Fill each chicken breast with 2 Tbsp. of the spinach mixture.
Set the griddle over medium-high heat and warm the remaining 1 Tbsp. olive oil.
Place the chicken, skin side down, on the griddle and cook until browned underneath, about 5 minutes. Turn the chicken over and cook until browned on the other side, about 3 minutes.
Transfer the chicken, skin side up, to the rack and set the rack on top of the griddle. Transfer the griddle to the oven and bake until an instant-read thermometer inserted into the thickest part of the chicken registers 160°F/71°C, about 20 to 22 minutes.
Let the chicken rest for 5 minutes before serving.