Prep the oven. Preheat the oven to 375°F and line a baking sheet with parchment paper.
Cream the butter and sugars. In a large bowl, beat the butter, brown sugar, and granulated sugar on high speed until light and fluffy, about 2 minutes. Scrape down the bowl as needed.
Add eggs and vanilla. Beat in the eggs one at a time, then mix in the vanilla extract.
Mix dry ingredients. Add the flour, baking soda, and salt and mix just until combined.
Stuff the cookies. Scoop rounded tablespoons of dough. Press a Milk Dud into the center, wrap the dough around it, and roll into a ball. Place about 8 cookies per baking sheet.
Top and bake. Sprinkle with flaky sea salt if desired. Bake for about 10 minutes, until the tops are just lightly browned.
Cool. Let cookies cool on the baking sheet for 5 minutes before transferring to a wire rack.
Notes
Dough texture: If dough is too soft, chill for 1 to 2 hours before scooping.
Cooling: Allow cookies to cool slightly so the melted Milk Duds do not stick or tear the cookies.
Reheating: Microwave cookies for about 15 seconds for a warm, gooey center.
Storage: Store in an airtight container at room temperature for 3 to 5 days.
Freezing: Freeze baked cookies for up to 2 months. Flash freeze first, then transfer to a freezer safe container.