In a 4-qt saucepan or similar size soup pot, combine the onion and oil. Cook over med-high heat, stirring frequently, 3-4 minutes, until onion is soft and beginning to brown.
Stir in the broth, sweet potatoes, ginger root, thyme, allspice, and crushed red pepper. Bring to a boil over high heat. Reduce the heat; simmer, uncovered, stirring occasionally, until the sweet potatoes are just tender, 7-8 minutes.
Scoop up 3 1/2 cup vegetables, then 1 cup broth from pan and place in a food processor. Process until completely pureed and smooth. Return the pureed mixture to the saucepan.
Combine about 2/3 of the canned tomatoes and the peanut butter in the processor. Process until well blended. Return the pureed mixture, then the remaining tomatoes, to the saucepan.
Bring the mixture to a simmer. Cook 2-3 minute longer, until the bisque is piping hot and the flavors are blended. Taste and add salt and pepper, if desired.