In a medium bowl, whisk together the flour, baking soda, kosher salt, ginger, cinnamon, cloves, and nutmeg. Stir in the old-fashioned oats until evenly distributed. Set the mixture aside.

In a large mixing bowl, beat the softened butter, dark brown sugar, and granulated sugar on medium speed for 2–3 minutes. The mixture should look creamy and slightly lighter in color. Scrape down the sides of the bowl as needed.

Add the molasses and beat until evenly incorporated: Mix in the egg and vanilla extract. The mixture should look smooth, glossy, and deep brown.

Add the flour-oat mixture to the wet ingredients in two additions. Mix on low speed just until no dry flour remains. The dough should be thick, sticky, and heavily textured with oats. Avoid overmixing.

Cover the bowl and refrigerate the dough for 30 minutes. After chilling, it should be noticeably firmer and easier to scoop.

Preheat the oven to 350°F (175°C) . Line two large baking sheets with parchment paper.

Scoop the chilled dough into 24 portions using about 1½ tablespoons per cookie. Arrange 12 portions on each prepared baking sheet, leaving about 2 inches between them. Gently roll rough portions into balls if needed, but do not flatten them.

Bake one sheet at a time for 10–12 minutes . The edges should look set and lightly browned, while the centers should still appear soft and slightly underdone. The cookies will continue setting as they cool.

Bake the second sheet in the same way: If the unbaked dough becomes very soft while waiting, return it to the refrigerator until the oven is available.

Let each batch cool on the baking sheet for 5 minutes. Transfer the cookies to a wire rack and cool for another 20 minutes. They should be completely cool before icing.

In a small bowl, whisk together the powdered sugar, whole milk, vanilla extract, and fine salt. The icing should be smooth and thick enough to drizzle while still flowing slowly from a spoon. If it is too thick, add milk a few drops at a time. If it becomes too thin, whisk in a little more powdered sugar.

Drizzle the icing over the completely cooled cookies with a spoon or piping bag. Leave some of the cookie surface visible rather than covering each cookie completely.

Arrange the iced cookies in a single layer and let them stand for about 25 minutes. The icing should feel dry to a light touch before the cookies are stacked or stored.