Adjust oven racks to upper and lower middle positions, and heat oven to 350 degrees F.
Line two large cookie sheets with parchment paper.
Whisk flour, baking soda, and salt in medium bowl; set aside.
Zest lemons with microplane; set aside.
In the bowl of an electric mixer, beat butter, 1 cup granulated sugar, lemon zest and lemon juice at medium speed until light and fluffy, about 3 minutes, scraping down sides of bowl as needed Add egg; beat at medium speed until combined. Add dry ingredients and beat at low speed until just combined, scraping down bowl as needed.
Cover bowl with plastic wrap; chill for an hour.
Roll half ounce balls of dough and roll them in remaining quarter cup sugar. Space balls about 2 inches apart on baking sheet. Press gently with the bottom of a water glass to a half inch thickness.
Bake until cookies are golden brown around the edges and just set and very lightly colored in the center, approx. 9 minutes, reversing position of cookie sheets from front to back and top to bottom halfway through baking time.
Remove from oven and cool on baking sheet about 5 minutes at room temperature. Transfer cookies to wire rack and cool to room temperature.
10. While cookies are cooling, combine cream cheese and lemon juice in a small bowl. Add confectioners’ sugar and mix until smooth.
11. Spoon a half tsp. glaze onto completely cooled cookies and gently spread with the back of a spoon. Allow glaze to harden several hours at room temperature for best results.