For the cookies: Preheat the oven to 325 degrees F.
Place the butter, powdered sugar and vanilla in the bowl of an electric mixer equipped with a paddle attachment (the one that looks like a peace sign). Beat the butter, sugar and vanilla together until creamy and homogeneous, 3 to 4 minutes. Remove the bowl from the mixer and, using a rubber spatula, fold in the flour and salt. Try to do this briskly without mixing the batter too much-mix until the flour is just combined.
Working in batches, transfer the batter to a pastry bag fitted with a large star tip. Pipe the dough onto an ungreased baking sheet in 2-inch circles. Bake the cookies for 12 to 15 minutes or until golden. Let them cool completely.
For the Caramel Dipper: In a large saucepan, combine the sugar, lemon juice and 1/4 cup water. Bring the pot to a boil (BTB). Be careful not to swish the pan around too much at this point because it can cause the sugar to recrystallize.
As the sugar mixture boils, the water will evaporate and the sugar will begin to turn brown, or caramelize. Be patient, this will take some time, 12 to 15 minutes. DO NOT take your eyes off the sugar! The sauce can go from golden to ruined very quickly. Relax and enjoy it-you're living on the edge and that's exciting!
Gradually the sugar will begin to turn gold, then light brown and then one shade past light brown-this is when you want to remove the pan from the heat and immediately add the heavy cream. As you do this, the mixture will bubble up like crazy. Whisk the cream until things settle down, then add the butter, two pats at a time. Be sure the butter is completely incorporated before adding more.
Once all the butter has been added, let the caramel cool. To serve, put 4 or 5 cookies on each plate with a little dipping dish of about 1/4 cup of the sauce. Sprinkle the sauce with a few grains of rock salt.