Preheat a grill to medium. Prepare the chicken: With your knife parallel to the cutting board, slice each chicken breast in half horizontally without cutting all the way through; open like a book. Cover with plastic wrap and pound until 1/2 inch thick.
Make the filling: Mash the garlic-and-herb cheese, artichoke heart, parsley, dill and lemon zest in a bowl. Whisk the lemon juice and olive oil in a separate bowl; season with salt and pepper.
Season the pounded chicken on both sides with salt and pepper and brush with half of the lemon oil; top with the spinach. Spoon the filling over the top, leaving a 1-inch border around the edges. Roll up the chicken toward the pointy end; tie with 3 pieces of kitchen twine. Brush with the remaining lemon oil.
Brush the grill grates with olive oil. Grill the chicken breasts, turning, until cooked through, 18 to 20 minutes. Transfer to a cutting board and let rest 5 minutes, then remove the twine and slice.