Using a wooden spoon, cream the butter until it's smooth (see tip below.) Stir in the sugars and vanilla extract until they're evenly blended. In a separate bowl, whisk together the flour, salt, and baking powder. Add the dry ingredients to the creamed mixture one third at a time, stirring after each addition, until the dough is evenly blended.
Set aside half of the cookie dough. Add the cocoa powder to the remaining dough and stir until it is fully incorporated.
Divide the chocolate dough in half. Place each half on a large piece of plastic wrap, pat it into a 1/2-inch-thick square, and then wrap it in the plastic. Repeat the process for the vanilla dough. Refrigerate the 4 squares until firm, about 1 to 2 hours.
Unwrap one piece of the chilled vanilla dough, leaving it on the plastic. Cover it with a second piece of plastic wrap, then roll it into a rectangle measuring about 9 by 7 inches. If the dough gets too soft and difficult to roll, slide it onto a baking sheet and refrigerate it until it's firm again, about 5 to 10 minutes. Unwrap a piece of the chocolate dough, cover it with another piece of plastic, and roll it into an 8- by 7-inch rectangle.
Remove the top piece of plastic wrap from both rolled doughs and invert the chocolate dough on top of the vanilla dough, lining up 3 sides and leaving about 1 inch of vanilla dough exposed on one side. Starting at the opposite side, lift the vanilla dough by the plastic wrap underneath it and snugly roll up both layers into a log, peeling away the plastic wrap as you go. Wrap the log in the plastic and refrigerate it for at least 2 hours. Repeat the process with the 2 other pieces of dough.
Heat the oven to 350°F Line 2 shiny, heavy baking sheets with parchment paper. Unwrap one of the logs and slice it into 1/4-inch-thick rounds. (See tip below for a safe, kid-friendly way to slice dough.) Arrange the slices on the parchment about 1 inch apart and bake them for about 13 minutes. When done, they'll still be a little soft to the touch -- the cookies will have a better texture if you don't let their edges brown.
Leave the cookies on the sheet for 1 minute, then transfer them to a wire rack to cool. Slice and bake the second log as the first.
A Kid-friendly Way to Slice Dough Slip a long piece of unflavored dental floss (about 20 inches) beneath the dough log. Pull the ends up and over the log, crossing them. Wrap the ends of the floss around your fingers and pull in opposite directions.