Mix the butter, tarragon, parsley, chives, thyme, 1 teaspoon salt and 1/4 teaspoon pepper in a bowl. Scoop the mixture onto a piece of plastic wrap or parchment paper and roll into a 1-inch-thick log. Wrap and freeze until firm, at least 3 hours and up to 3 days.
Slice the frozen butter into 6 equal pieces. Lay a chicken breast between 2 pieces of plastic wrap or 2 large resealable plastic bags; pound to 1/4 inch thick with a heavy skillet or mallet. Transfer to a clean piece of plastic wrap and season with salt and pepper. Put a piece of herbed butter in the center of the chicken; using the plastic wrap as an aid, tightly roll up the chicken starting at a long side, tucking in the short sides as you go. Tightly wrap the chicken roll in plastic wrap. Repeat with the remaining chicken and butter. Refrigerate the rolls at least 2 hours or overnight.
Preheat the oven to 350 degrees F and set a wire rack on a rimmed baking sheet. Put the flour in a shallow dish. Whisk the eggs and mustard in another shallow dish. Put the panko in a third dish. Unwrap the chicken rolls and generously season with salt and pepper. One at a time, dredge the chicken rolls in the flour, then dip in the egg mixture and roll in the panko to coat.
Heat 1/2 inch of vegetable oil in a large skillet over medium-high heat. Fry the chicken rolls in 2 batches, turning, until golden, 6 to 8 minutes. Transfer to the prepared rack and bake until cooked through, about 10 minutes.