Line a small plate or tray with parchment paper. Scoop 12 small balls of nutella (about 2 teaspoons each) onto the tray and freeze for at least 1 hour until solid.
Beat the softened butter, dark brown sugar, and granulated sugar in a large bowl with an electric mixer on medium speed until light and fluffy, about 3 minutes.
Add the egg, egg yolk, and vanilla extract. Beat on medium speed until smooth and well combined, about 1 minute.
In a separate bowl, whisk together the flour, cocoa powder, baking soda, and salt. Add the dry ingredients to the wet mixture and stir with a spatula until just combined. Fold in the chocolate chunks. Refrigerate the dough for 30 minutes.
Preheat your oven to 350°F (177°C) and line baking sheets with parchment paper. Scoop about 2 tablespoons of dough, flatten it in your palm, place a frozen nutella ball in the center, then wrap the dough around it and roll into a smooth ball. Place on the prepared baking sheet with 3 inches of space between each cookie.
Bake for 10 to 12 minutes until the edges are set but the centers still look soft and slightly underdone. Sprinkle flaky sea salt on top immediately.
Let the cookies cool on the baking sheet for 10 minutes. The centers will be gloriously molten when you break them open. Serve warm for the best experience.