Using a paring knife, make an incision in the side of each chicken breast, passing the knife though horizontally to make a slit but not cutting through the other side of the breast. With the paring knife inserted, rotate the knife to carve out a bit more space and create a pocket. Season the outside and inside of the chicken with salt and pepper.
Lightly beat the egg white in a bowl to break it up, then add the goat cheese, parsley, tarragon, shallots, mustard and lemon zest and mix until incorporated. Transfer the mixture to a resealable plastic bag (or piping bag). Snip off a corner of the plastic bag and pipe the mixture into each chicken breast until filled.
Heat the oil in a large ovenproof saute pan over medium-high heat until hot and glistening. Add the chicken breasts skin-side down and cook until the skin turns a dark golden brown, 8 to 10 minutes. Add the butter, garlic and thyme, then flip and transfer the pan to the oven; cook the chicken until the internal temperature reaches 160 degrees F, about 10 minutes. Allow the chicken to rest for 5 minutes before slicing and serving.