Combine 1/2 cup water, 1/3 of the onion, ginger, and jalapeno in a blender; puree until smooth.
Place goat meat in a large pot. Pour onion puree on top. Stir in 1 cup water, shrimp cube, and salt. Cover and cook over medium-high heat, stirring occasionally, until flavors combine, about 20 minutes. Reduce heat and add remaining 2/3 of the onion; cook, covered, until softened, about 20 minutes.
Transfer 2/3 of the onion to the blender. Add 1 cup water; puree until smooth. Pour back into the pot.
Mix 1 cup water, peanut butter, and tomato paste in a saucepan over medium-low heat. Cook, stirring often to prevent scorching, until oil separates, 15 to 20 minutes. Remove from heat. Stir in 1 1/2 cup water with a fork. Pour into the pot.
Stir basil leaves into the pot. Bring soup to a boil; reduce heat and simmer until peanut oil rises to the surface, 20 to 25 minutes.