
In a small bowl, combine pepper, cumin, cardamom, and 2 tsp. salt. In a medium bowl, combine 2 1/2 tsp. seasoning mix, half of ginger, half of garlic, and 1 tbsp oil. Pat chicken dry with paper towels and add to bowl, tossing to coat. Let chicken sit at room temperature at least 15 minutes or refrigerated up to 2 hours.
In a large Dutch oven or pot over medium heat, heat remaining 2 tbsp. oil. Arrange chicken in a single layer and cook, undisturbed, until a golden brown crust starts to form, 5 to 6 minutes per side. Transfer chicken to a plate.
In same pot, cook onion, carrot, and remaining seasoning mix, stirring occasionally and scraping up browned bits in bottom of pot, until softened, about 5 minutes. Add pepper and remaining ginger and garlic and cook, stirring, until fragrant, about 1 minute. Add thyme and tomato paste and cook, stirring frequently, until paste darkens in color, about 4 minutes. Add chopped tomatoes and 2 tsp. salt and cook, stirring occasionally and breaking up tomatoes with a wooden spoon, until mostly broken down, about 20 minutes.
In a large bowl or measuring cup, whisk peanut butter and 1 cup broth until smooth. Add remaining broth 1 cup at a time, whisking until smooth and combined. Add broth mixture and potatoes to pot, stirring to combine. Reduce heat to medium-low and simmer, stirring occasionally, until potatoes are tender and soup is thickened, about 30 minutes.
Chop chicken into bite-size pieces. Remove pepper from pot, then return chicken to pot. Add fish sauce and stir until chicken is heated through.
Divide rice among bowls. Ladle soup over rice and top with peanuts and thyme.
Make Ahead: Stew can be made 5 days ahead. Transfer to an airtight container and chill, or freeze up to 3 months.