Cook quick brown rice according to package directions; meanwhile, heat coconut oil in a large pot and stir in curry paste until fragrant (30 seconds).
Add red lentils, vegetable broth, and coconut milk to the pot; bring to a simmer and cook until lentils break down and soup thickens (18 minutes).
Stir in fresh spinach until wilted (2 minutes), then juice the lime and stir in the juice.
Divide cooked rice among four bowls and ladle curry lentil soup over top.