Start by preheating your oven to 350°F. While it heats, finely chop your walnuts into about 1/4-inch pieces and set aside in a shallow bowl for rolling. Measure out all your dry ingredients (flour and salt) into one bowl, and have your butter, brown sugar, egg, and vanilla at room temperature—this is crucial for proper dough texture. I like to use room temperature ingredients because they incorporate more smoothly and create a more uniform cookie.
In a large mixing bowl, cream together the softened butter and packed brown sugar until light and fluffy, about 2-3 minutes. Add the vanilla extract and mix well, then beat in the egg until fully incorporated. Gradually add the dry ingredient mixture from Step 1 to the wet ingredients, mixing on low speed until just combined—don't overmix, as this will make tough cookies. The dough should come together into a cohesive ball.
Working with the dough from Step 2, roll it into 1-inch balls using your hands—work quickly so the dough doesn't warm up too much. Immediately roll each ball in the chopped walnuts from Step 1, pressing gently so the nuts adhere well. Place the coated balls about 2 inches apart on parchment-lined baking sheets. Once all cookies are formed, use your thumb to press a deep indent into the center of each cookie, creating a well about the size of a dime—this is where your chocolate will go.
While the cookies are on the baking sheet ready to bake, heat 1/3 cup light corn syrup in a microwave-safe bowl for about 1 minute until warm and slightly thickened. Add the semisweet chocolate chips to the warm syrup and let sit for 30 seconds, then stir until completely melted and smooth. If using a pastry bag, transfer the chocolate mixture to the bag fitted with a small round tip, or you can use a small spoon to fill the cookies. I find the pastry bag gives you better control for filling the indents evenly.
Pipe or spoon the melted chocolate mixture from Step 4 into the thumb indents of each cookie on the baking sheet, filling each well generously but not overflowing. Place the baking sheets immediately into your preheated 350°F oven and bake for 10-12 minutes, until the cookies are set and firm but still slightly soft to the touch. The chocolate will set as the cookies cool, so don't overbake.
Remove the baking sheets from the oven and let the cookies cool on the sheet for 2-3 minutes to firm up slightly. Transfer the cookies to a wire cooling rack and allow them to cool completely—the chocolate centers will harden as they cool. Once cooled, the cookies are ready to enjoy or store in an airtight container for up to 5 days.