In a large bowl with an electric hand mixer or in a stand mixer fitted with a paddle attachment, beat melted butter, brown sugar, salt, and ½ cup of granulated sugar on medium speed until smooth and slightly lightened in color, about 1 minute.

Add egg, vanilla, molasses, and 2 tablespoons instant espresso powder, and beat on low speed until smooth, about 30 seconds.

In a separate medium bowl, whisk flour, ginger, baking soda, cinnamon, and cream of tartar until evenly combined.

Reduce mixer speed to low; gradually add flour mixture to butter mixture, beating just until no dry streaks remain, about 30 seconds.

Cover bowl and refrigerate for 30 minutes to allow dough to firm slightly.

Adjust oven rack to middle position and preheat oven to 375°F (190°C). Line 3 rimmed baking sheets with parchment paper.

In a shallow bowl, stir together remaining ¼ cup sugar and remaining 2 teaspoons espresso powder until combined.

Using a #40 scoop, scoop 8 portions of dough (30 g each, about 2 tablespoons), and roll into balls.

Roll and coat each ball in sugar mixture, and arrange 2 to 3 inches apart on prepared parchment-lined baking sheet.

Keep remaining dough covered and chilled until ready to portion and bake next sheet.

Bake until cookies are puffed and set around the edges, 8 to 10 minutes.

Remove from oven and tap baking sheet firmly on counter once to deflate centers for chewiness.

Cool 5 minutes on baking sheet, then use a spatula to transfer cookies to a wire rack to cool completely, about 30 minutes.

Repeat scooping, rolling, and coating process with remaining dough, arranging 8 portions per baking sheet for a total of about 24 cookies.