Beat butter at medium speed of an electric mixer until creamy. Gradually add sugars, beating until fluffy. Add eggs, one at a time, beating after each addition. Add sweet potato and vanilla; beat until blended.
Combine flour, baking powder, and salt; add to butter mixture, beating at low speed until blended. Turn dough out onto a lightly floured surface, and knead in pecans. Divide dough in half. Using floured hands, shape each portion into a 10" x 6" log on a lightly greased cookie sheet.
Bake at 375° for 25 minutes. Cool 5 minutes on cookie sheet; remove to a wire rack, and cool to the touch. Reduce oven temperature to 325°.
Cut each log crosswise into 1/2"-thick slices with a serrated knife. Place slices, cut side down, on ungreased cookie sheets.
Bake at 325° for 15 minutes; turn slices over and bake 15 more minutes. Turn slices upright, and return to oven. Turn oven off, and leave in oven 15 minutes or until dry. Cool completely on wire racks.
Note: You can sweeten the flavor or change the length of this biscotti.
For a sweeter treat, dip biscotti in vanilla-flavored candy coating. Here's how: Melt coating, 8 ounces at a time, in an 11" x 7" baking dish in the microwave according to the package directions. Dip each biscotti, cut side down, into melted coating. Place biscotti, dipped side up, on wax paper until coating is hardened.
For long biscotti, carefully cut each log lengthwise into 1/2"-thick slices after the first baking. Yield: 1 1/2 dozen.