Line a baking sheet with parchment paper. Scoop 1-tablespoon balls of Nutella onto prepared baking sheet (24 balls total). Transfer baking sheet into the freezer and allow the Nutella balls to freeze completely, about 30 minutes.
In a large bowl using a hand mixer, beat butter and both sugars until light and fluffy. Add eggs, milk, and vanilla and beat until combined, then add flour, cocoa powder, baking soda, and salt.
Place dough in the freezer, along with the Nutella, is and let set for 15 minutes.
Preheat oven to 350° and line a baking sheet with parchment paper.
Scoop a heaping tablespoon of cookie dough and flatten into a pancake-like circle. Top with a frozen Nutella ball and cover with dough, pinching to seal and adding more dough if necessary to completely cover. Transfer to prepared baking sheet and repeat with remaining dough and frozen Nutella, spacing cookies 2 inches apart.
Sprinkle cookies with flaky sea salt and bake until puffed, 15 minutes. Let cool on pan 5 minutes before serving.