Preheat oven to 350 degrees, with racks in upper and lower thirds. Line two baking sheets with parchment paper; set aside. Whisk together flour, baking soda, cornmeal, and salt in a medium bowl.
Put butter and sugar in the bowl of an electric mixer fitted with the paddle attachment; cream on medium speed until pale and fluffy, about 3 minutes. Mix in eggs, 1 at a time. Add flour mixture; mix on low speed until just combined. Mix in currants and thyme.
Using a tablespoon or a 1 1/2-inch ice-cream scoop, drop rounded balls of dough onto lined sheets, spacing them 2 inches apart. Bake, rotating and switching positions of sheets about halfway through, until pale golden, 10 to 12 minutes. Transfer cookies, on parchment, to a wire rack. Let cool completely.