Heat oil in a large nonstick skillet over medium-high heat. Add onion; sauté 2 minutes or until soft. Add tempeh; sauté 6 minutes or until beginning to brown.
Combine tomato paste, vinegar, brown sugar, paprika, 3/4 teaspoon salt, and 1/4 teaspoon pepper in a small bowl. Add tomato mixture to tempeh; stir to combine. Stir in 3 cups water. Simmer until mixture is thick and liquid is almost evaporated (about 20 minutes), stirring often. Remove from heat; keep warm.
Combine mayonnaise, 2 teaspoons water, remaining 1/4 teaspoon salt, and remaining 1/4 teaspoon pepper in a medium bowl. Add shredded cabbage and carrot, tossing well to combine.
Top each of 6 bottom halves of hamburger buns with about 3/4 cup tempeh filling. Top each serving with about 1/3 cup cabbage mixture, and cover with top half of each bun. Serve sloppy joes immediately.