PREHEAT oven to 350F. Line 2 baking sheets with parchment.
WHISK flour, cocoa, baking soda and salt in a bowl.
BEAT butter and both sugars in a large bowl using an electric mixer on medium until fluffy, about 3 min. Add egg and beat for 1 min, scraping sides and bottom of bowl. Add Nutella and vanilla and beat until smooth. Beat in flour mixture on low until just combined.
SCOOP out 2-tsp portions of dough and roll into balls. Place 16 cookies 1 in. apart on a prepared sheet. Bake until puffed and slightly cracked on top, about 5 min. Let cool for 1 min, then transfer to a rack to cool. Repeat with remaining dough.
FROSTING: Beat butter and Nutella in a large bowl with an electric mixer on medium-high until fluffy, 2 min. Add icing sugar and salt and beat on medium-low until well combined. Increase speed to medium and beat until light, 1 min. Add milk and beat for 2 min. Transfer to a piping bag fitted with a small star tip.
FLIP half of the cookies and pipe about 1 tbsp frosting onto each, then top with remaining cookies.