Line a cookie sheet (or plate) with baking paper. Drop small dollops of Nutella - about 1 1/2 teaspoons each - onto the lined cookie sheet. Place in the freezer as you make the dough.
Preheat the oven to 350F (180C). Line cookie sheets with parchment paper or silicone baking mats.
In a large bowl, beat together the butter, brown sugar, granulated sugar and peanut butter until smooth (you will still see sugar granules, but there shouldn't be lumps of sugar or butter).
Mix the egg and vanilla extract into the butter mixture.
Add in the flour, baking soda and salt. Start with the mixer on a low speed and gradually mix together until incorporated. You should no longer see lumps or streaks of dry ingredients. Turn off the mixer and gently squeeze the dough between your fingers. If it leaves globs of dough attached to your fingers, mix in 1-2 more tablespoons of dough.
Scoop out about 1 to 1 1/2 tablespoons of dough (I use a cookie scoop for this). Flatten slightly and place on the lined cookie sheet. Take one frozen Nutella dollop out of the freezer and place in the middle of the dough. Pinch the sides of the dough around the dollop of Nutella. As needed, scoop out another 1-2 teaspoons of dough to wrap around the Nutella dollop to ensure that the Nutella dollop is fully enclosed in cookie dough with just the top peaking out. Repeat the process with the rest of the dough. Keep the Nutella dollops in the freezer, removing 1 at a time so that they stay frozen for when you assemble the cookie dough balls. Place cookie dough balls about 2 inches apart on the lined cookie sheets.
Bake the cookie sheets one at a time in the middle of the preheated oven. They'll bake for about 9-12 minutes, or until the tops look set.
Cool on the cookie sheets for at least 15 minutes before transferring to a wire rack to continue cooling.