Heat a medium saucepan over medium heat. Add the butter and let it cook, swirling occasionally, until the milk solids turn golden brown and the butter foams and begins to smell nutty. Remove from the heat and add the split vanilla bean along with the scraped seeds. Set aside until cooled to room temperature, then remove the vanilla bean halves.

Place the cooled brown butter in the bowl of a stand mixer fitted with the paddle attachment and add the brown sugar and granulated sugar and  cream the butter-sugar mixture until pale and fluffy.

Add the egg, then the egg yolk and mix, scraping the sides of the bowl as necessary.

Turn off the mixer and add the flour, baking soda and salt. Mix on very low speed just until the dough begins to come together. Drizzle in the milk, then add the chocolate and mix just until the dough comes together. Wrap the dough in plastic wrap and refrigerate for at least 1 hour.

Preheat the oven to 350º F.

Portion the dough into 2-ounce balls and roll between your hands until smooth. Place the cookie dough onto parchment-lined baking sheets, spacing cookies at least 3 inches apart. Bake in the center of the oven for 7 minutes, then remove from the oven and sprinkle the top of each cookie lightly with the fleur de sel. Return the cookies to the oven until they are lightly golden around the edges but still soft to the touch, 5 to 6 minutes longer. Remove from the oven and place the baking sheets on wire racks to cool briefly before transferring the cookies to a wire rack.

To make the Vanilla Milk:

Whisk the milk, sugar, and vanilla beans together. Strain, then chill.

Serve the cookies warm with a small glass of the chilled Vanilla Milk.