Preheat oven to 400°. Scoop out squash seeds; discard or reserve for another use. Place squash, cut-side down, on a baking sheet coated with cooking spray. Bake 40 minutes or until squash is tender. When cool enough to handle, scoop out pulp. Process squash pulp in a food processor until smooth. Turn oven down to 375°. Coat a 13- x 9-inch baking pan with cooking spray, and line with parchment paper.
Combine flour and next 7 ingredients in a medium bowl. In a large bowl, beat butter and brown sugar at medium speed with an electric mixer until creamy. Add eggs; beat 1 minute. Add tamarind nectar and 1 1/3 cups butternut squash puree; beat until smooth. Add flour mixture, beating until combined. Pour batter into prepared pan, and bake at 375° for 30 minutes or until a wooden pick inserted in center comes out clean.
While cake is baking, prepare Mango Compote. Beat yogurt and powdered sugar with an electric mixer until creamy.
Cut cake into squares. Cut each square into two triangles, and top with yogurt sauce. Serve with Mango Compote with Cilantro-Mint Syrup.