Prepare the cookies: Beat butter, brown sugar, molasses, vanilla, and egg with an electric mixer on medium speed until light and fluffy, about 2 minutes.
Whisk together flour, graham cracker crumbs, salt, baking powder, baking soda, and cinnamon in a large bowl. Add flour mixture to butter mixture alternately with cream, beginning and ending with flour mixture. Beat on low speed just until mixture comes together after each addition. Cover dough, and chill 30 minutes.
Preheat oven to 350°F. Scoop dough, using a 1 1/2-inch cookie scoop, onto 2 parchment paper-lined baking sheets, placing cookies about 2 1/2 inches apart. Press dough to flatten slightly. Bake in preheated oven until golden and set, about 13 minutes, rotating pans top rack to bottom rack halfway through baking time. Cool completely on pans on wire racks.
Meanwhile, prepare the filling: Beat butter with an electric mixer fitted with paddle attachment on medium speed until smooth, about 1 minute. Reduce speed to low, and slowly add powdered sugar. When sugar is fully incorporated (after about 3 minutes), increase speed to medium-high. Add marshmallow crème, and beat until smooth, about 2 minutes. Spoon into a gallon ziplock plastic bag. Set aside.
Prepare the glaze: Place chocolate in a medium-size microwavable bowl; stir in oil. Microwave on HIGH until melted and smooth, 1 1/2 to 2 minutes, stirring every 30 seconds. Let stand 5 minutes, stirring occasionally.
Place half of the cookies, bottom sides up, on a wire rack set in a rimmed baking sheet. Cut a small hole in the corner of the ziplock bag, and pipe 2 to 3 tablespoons of filling onto each cookie. Top with remaining cookies, bottom side down, and gently press until filling meets edge of cookies. Carefully spoon 2 to 3 tablespoons of glaze over each cookie, allowing glaze to drip down sides. Use a small offset spatula to spread glaze evenly over top and around the sides. Sprinkle with sea salt, and chill until chocolate is set, about 20 minutes. Keep chilled until ready to serve.