In a large Dutch oven, warm 3 tbsp olive oil over medium heat. Add onion and leek; cook 6–8 minutes until translucent. Add garlic; cook 1 minute.
Stir in tomato paste; cook 1 minute. Add dried spearmint, dill, kosher salt, black pepper, and optional red pepper flakes; stir 30 seconds.
Add the brown rice; stir to coat. Pour in broth and bring to a gentle boil.
Reduce heat to low, cover, and simmer 25 minutes.
Uncover; add spinach in batches, stirring to wilt. Cover and simmer 10–15 minutes, until rice is tender and most liquid is absorbed.
Off heat, stir in lemon juice and zest. Drizzle remaining 2 tbsp olive oil, fluff, rest 5 minutes, and garnish with dill.