Make lemon curd at least a day in advance. Chill for at least 8 hours.
Beat butter and 3/4 cup sugar in a large bowl with a hand mixer or a stand mixer fitted with the whisk attachment until smooth and creamy.
Mix in vanilla, cornstarch, and salt. Add flour and beat on low until just combined, scraping the sides of the bowl as needed. Avoid overmixing.
Roll dough into 1-tablespoon-sized balls. Roll in 1/3 cup sugar, then press a small indent into the center of each ball.
Arrange on a lined sheet pan, cover, and refrigerate for 1 hour (or freeze for 30 minutes).
Fill indents with chilled lemon curd (careful to not overfill at all), then refrigerate for 10 more minutes.
Preheat oven to 350°F. Bake cookies for 8–10 minutes, until edges are very lightly browned and tops are set.
Cool cookies completely on a wire rack. Repeat with remaining cookies.
Whisk together glaze ingredients in a bowl, adjusting cream/milk for desired thickness.
Transfer to a plastic bag and cut off the tip. Drizzle over cookies and allow to set.
