Preheat oven to 350°F. Line two large baking sheets with parchment.
Beat butter until smooth. Add both sugars and beat on medium-high until fluffy. Beat in egg, egg yolk, and vanilla on high. Beat in ¼ cup Nutella until combined.
Whisk flour, cornstarch, baking soda, and salt. Mix into wet ingredients on low. Add 1 cup chocolate chips and mix 5–10 seconds. Turn off mixer. Add remaining ¼ cup Nutella and beat just 5–10 seconds so it streaks but doesn't fully incorporate — dough will be slightly crumbly.
Scoop and roll dough into 1-tbsp balls. Bake 10 minutes until slightly golden at the edges. The centers will look very soft.
Cool 5 minutes on the sheet. Press remaining chocolate chips into the warm tops and sprinkle with sea salt. Transfer to a rack to cool completely.