Make cookies: In a medium bowl, whisk together flour, ginger, cinnamon, baking soda, and salt. Set aside. Using a stand mixer with paddle attachment, beat butter and brown sugar on medium-high speed until light and fluffy, 3 minutes. Reduce speed to low and add molasses, then egg and vanilla, mixing until just combined. Add dry ingredients and mix until a smooth dough forms. Chill airtight until firm, at least 2 hours.
Meanwhile, make lime curd: In a medium metal bowl, whisk together egg yolks, lime juice, and granulated sugar. Set over a saucepan of 1 in. simmering water (bowl should not touch water) and cook, whisking constantly, until mixture thickens and reaches 165° to 170° on an instant-read thermometer, 5 to 10 minutes. Remove bowl from pan and whisk in butter and lime zest. Let cool slightly, then press plastic wrap onto surface of lime curd. Chill until cold, 1 hour.
Preheat oven to 375° and line 2 rimmed baking sheets with parchment paper. Roll dough into 1-in. balls, then roll in granulated sugar to coat. Transfer to lined sheets, spacing about 11/2 in. apart. With handle of a wooden spoon, make an indentation 1/2 in. wide in center of each ball of dough, pressing halfway down. (These “thumbprints” will spread in the oven.)
Bake cookies until slightly darker, 10 to 12 minutes. Remove from oven and reinforce indentations using wooden spoon handle. Let cool on pans a few minutes, then transfer to wire racks to cool completely.
Fill each cookie with 1 tsp. lime curd and sprinkle with lime zest.
MAKE AHEAD Dough, up to 1 day, chilled airtight; or 1 month, frozen; lime curd, up to 2 days, chilled airtight. Filled cookies, up to 2 days, chilled airtight in a single layer (they’ll soften a little); or 1 month, frozen (thaw in a single layer).