To prepare cookie dough, sift the flour and salt in a large bowl. Mix in walnuts. Set aside.
Cream butter and powdered sugar in a bowl of a standing mixer until light and fluffy. Beat in vanilla.
Reduce the speed to low and beat in the flour mixture until just incorporated.
Gather dough and form a dish. Cover with plastic wrap and refrigerate for about 30 minutes to chill.
Preheat oven to 350F. Line 2 baking trays with parchment paper.
On a lightly floured surface, roll out the dough to 1/4 inch thickness. Cut into 3-inch rounds using a floured cookie cutter. Place on the prepared baking sheets and place in the refrigerator for about 15 minutes.
Bake the cookies for 8 to 10 minutes, or until cookies are lightly browned.
Remove from oven and let the cookies cool a few minutes on the baking sheet before transferring them to a wire rack to cool completely.
To prepare ice cream, combine milk, heavy cream, salt and dulce de leche in a medium sauce pan. Bring mixture to simmer, until dulce de leche is dissolved. Remove from heat.
Strain the mixture into another medium bowl. Let cool over a bowl of ice water, or chill in the refrigerator, stirring occasionally.
Process mixture in an ice cream maker according to manufacturer’s instructions. Transfer to a 9 x 13-inch baking pan, level the top, cover and freeze until firm, at least 4 hours.
To assemble the sandwiches, remove ice cream from the freezer, use a cookie cutter to cut out a disc of ice cream and place it on top of one cookie. Place another cookie on top of the ice cream. Lightly press to adhere the sandwich.
Roll or sprinkle the sides with chopped walnuts.
Serve immediately. Or wrap each sandwich with plastic wrap and place them in the freezer.