1. Preheat oven to 350°F (177°C).
2. Put the butter into the work bowl of a stand mixer. Cream the butter until light and fluffy.
3. Add the sugar and beat the butter and sugar mixture until well combined.
4. Add the eggs, one at a time, mixing well after each addition.
5. Add the lemon zest and vanilla. Mix until blended.
6. In a separate bowl, add the flour, salt, and baking soda. Stir or whisk until well blended.
7. Add the flour mixture to the batter.
8. Add the lemon juice and heavy cream. Mix on low speed until blended.
9. Remove work bowl from mixer. Cookie batter will be sticky.
10. Using a spatula, spoon the cookie dough onto the buttered jelly roll pan.
11. Dip your fingers into the extra flour and gently pat the dough so that it evenly covers the jelly roll pan.
12. Bake for 25 minutes or until the edges of the cookie are lightly browned and a cake tester comes out clean from the center of the pan.
13. Remove cookie from the oven and let it cool to room temperature in the pan.

For the Frosting

1. In a large, heavy-bottomed, high-sided pan, melt the butter, 1½ cups sugar, and heavy cream over medium heat.
2. When the butter is melted and the sugar dissolves, reduce the heat to low.
3. Add the salt and stir. Set aside on low.
4. Put another high-sided, large, heavy-bottomed saucepan on a burner. Turn the heat to medium-high.
5. When the pan is warmed, add 1 cup of granulated sugar. Shake the pan frequently.
6. When the sugar shows signs of melting at the edges of the pan, reduce the heat to medium.
7. Continue to shake the pan across the burner constantly until the sugar is melted and has turned a rich copper color.
8. Remove the caramelized sugar from the burner and carefully pour into the pan of melted butter, sugar, and cream mixture.
9. Stir the caramelized sugar into the cream mixture.
10. Using a candy thermometer, cook the caramel mixture until it just reaches the soft ball stage (235°F/113°C) over medium heat, stirring constantly.
11. Remove the caramel from the heat and cool slightly, 3 to 5 minutes.
12. Add the vanilla to the caramel and stir to incorporate.
13. Add one cup of sifted confectioners’ sugar and stir until blended.
14. Add another cup of the sifted sugar and stir.
15. Add the sour cream and stir until blended.
16. Add the final cup of sifted sugar and stir until blended.
17. Working quickly, ice the sheet of lemon sugar cookies.
18. Let icing cool.
19. Slice cookies and serve.