While the brown butter cools, prepare the Nutella centers for these cookies. Line a cookie sheet with parchment paper, and use two spoons to create 12 equal-sized scoops of Nutella using about 1 Tbsp of Nutella per scoop. Place this pan in the freezer to allow them to firm up.
Use a hand or stand mixer and paddle attachment to cream together the cooled brown butter (it should be partially solidified at this point and no longer warm to the touch) and 3/4 cup of packed light brown sugar for a couple of minutes, until the mixture lightens in color. Use a rubber spatula to scrape the sides and bottom of the bowl as needed.
Mix in 1 large egg, 1 Tbsp sour cream or Greek yogurt, and 1 tsp vanilla extract on a medium-high speed until the mixture is smooth.
Then add 1 1/4 cups of all-purpose flour, 1/2 tsp baking soda, and 1/2 tsp fine salt. Mix on low speed just until combined. Gently fold in 1 cup of roughly chopped chocolate chips.
The dough should be a bit sticky at this point. Use a cookie scoop to make 12 equal-sized cookie dough balls and place them on a parchment-lined sheet.
Chill the cookie dough balls for at least 30 minutes, or overnight. This also helps the dough absorb the brown butter flavor, hydrates the flour, and prevents the cookies from overspreading while they bake.
About 20 minutes before your cookies are done chilling, preheat the oven to 350 F / 175 C.
Once the dough is chilled, remove it and the chilled scoops of Nutella from the freezer.
Flatten each ball of cookie dough in the palm of your hand and place a frozen scoop of Nutella in the center. Fold the dough around the Nutella and shape it into a hockey puck, doing your best to fully cover the Nutella. Set aside and repeat with the remaining cookie dough. If the Nutella gets too soft to handle or the cookie dough starts sticking to your hands, pop them back into the freezer and/or fridge until they're easier to handle.
Then line two baking sheets with parchment paper or silicone baking mats.
Place the cookie dough balls 2-3 inches apart on the prepared cookie sheets. Top with a few extra bits of chocolate if desired.
Bake for 9-10 minutes, or until the edges just begin to brown. Rotate the pans halfway through to help the cookies bake evenly. Remove them from the oven and garnish with flakey sea salt while they're still hot.
Let the cookies cool on the pan for 15 minutes, then move to a wire rack to finish cooling. Enjoy with a glass of cold milk!