Finely chop walnuts and basil leaves. Place walnuts, basil,garlic,lemon zest,5 tbsp of butter and salt in bowl of an electric mixer. Using paddle attachment, mix on medium-high until butter is smooth.
Using a sharp thin-bladed knife, cut a large pocket in each of the chicken breast by inserting it horizontally into thickest and carefully slicing from side to side without piercing the edges of the breast.
Divide butter in 4 parts and use your fingers to stuff each pocket with a portion. Don't worry if it spills slightly; they will seem quite full. Season both sides of the chicken with salt and pepper.
Melt the remaining 1 tbsp. of butter in large skillet over medium heat. When it sizzles, lower chicken breasts into pan, cover lightly and reduce heat to low. Cook for 5 minutes or until nicely browned, then turn chicken over and continue to cook, covered, for 5 to 7 minutes longer or until there is no sign of translucency when you peek inside one of the pockets. An instant read thermometer inserted in the center of a breast should register 165 degrees. Some of the basil butter will have spilled out of the chicken to create a sauce in the pan. Transfer the chicken to a warm platter or individual serving plates and spoon the sauce over the top. Serve immediately