Heat 3 tbsp. oil in a medium pot over medium heat and add shallots. Cook, stirring occasionally, until softened and translucent, about 5 minutes. Add garlic and ginger and cook until aromatic, about 1 minute more.
Add chickpeas, broth, and salt. Cover and bring to a boil. Reduce to a simmer and cook until flavors are well blended and chickpeas are very soft, about 25 minutes. Add spinach and cook just until softened and bright green, about 2 minutes. Let cool 5 minutes.
Meanwhile, heat remaining 1 1/2 tbsp. oil in a small frying pan over medium-low heat. Stir in curry powder and cook until aromatic, 30 seconds. Remove from heat.
Whirl half of soup at a time in a blender, starting on low speed, until very smooth. Ladle into bowls. Just before serving, stir lime juice into curry oil and drizzle over soup. Garnish with mint.