Prepare the turkey: Stir together butter, thyme, garlic, rosemary, sage, lemon zest, 2 teaspoons of the salt, and 1 teaspoon of the pepper in a medium bowl until well incorporated. Rub butter mixture under turkey skin. Sprinkle outside of turkey with remaining 1 teaspoon salt and 1/2 teaspoon pepper. Pour chicken broth in bottom of a 6-quart slow cooker; add turkey, skin side up. Cover and cook on LOW until a thermometer inserted into thickest portion registers 165°F, 4 1/2 to 5 hours.

Remove turkey from the slow cooker; remove skin from turkey, and reserve for later use. Slice turkey and set aside, covered.

Prepare the Potatoes: Preheat oven to 375˚F. Toss potatoes with 2 Tbsp. olive oil, garlic, 3/4 tsp. kosher salt, and black pepper on a rimmed baking sheet. Bake at in preheated oven until tender, about 25 minutes.

Prepare the Sauce: Cook reserved turkey skin in 1 Tbsp. olive oil in a medium skillet over medium until it is browned and fat has rendered, about 5 minutes. Remove and discard turkey skin, reserving drippings in skillet. Add minced shallots. Cook until shallots are tender, about 2 minutes. Add white wine; cook until reduced by half, about 2 minutes. Add chicken broth, remaining 1/4 tsp. salt, and coarse-grain mustard, Dijon mustard, and chopped thyme. Simmer 2 minutes. Remove skillet from heat; stir in butter until melted. Serve sauce over sliced turkey and potatoes.