Stir together 2 cups flour, 1 1/4 tsp. cream of tartar, 3/4 tsp. baking soda, and 1/4 tsp. salt in a medium bowl. Stir together cocoa and remaining 1 2/3 cups flour, 1 1/4 tsp. cream of tartar, 3/4 tsp. baking soda, and 1/4 tsp. salt in another medium bowl.
Beat butter, vanilla, 2 cups sugar, and 1 tsp. cinnamon at medium speed with an electric mixer until creamy. Add eggs, 1 at a time, beating well after each addition. Spoon half of butter mixture into cocoa mixture; spoon remaining butter mixture into flour mixture.
Beat plain batter until just blended; beat chocolate batter until just blended. Chill both mixtures 30 minutes.
Preheat oven to 350°. Stir together remaining 1/3 cup sugar and 3 tsp. cinnamon in a small bowl. Drop plain dough by level tablespoonfuls onto aluminum foil. Top each with 1 Tbsp. chocolate dough; roll together into a ball. Roll in cinnamon-sugar; place 3 inches apart on parchment paper-lined baking sheets, and flatten.
Bake at 350° for 14 minutes or until edges are lightly browned, lightly tapping baking sheets halfway through to deflate cookies. Cool 5 minutes; transfer to wire racks, and cool.
Spread about 2 Tbsp. peanut butter onto half of cookies; top with remaining cookies.