Make soup: Sauté onion and ginger in oil in a medium pot over medium-high heat until golden, 5 minutes. Add garlic and coriander and cook until softened, 1 minute, then add pumpkin, broth, salt, and pepper. Simmer, covered, until pumpkin is very tender, 8 to 10 minutes. Purée in batches in a blender until very smooth.
Make pesto: Pound garlic, whole mint leaves, salt, and 1 tbsp. oil in a mortar into a coarse paste (or use a food processor). Add remaining oil and pumpkin seeds and pound until coarsely crushed.
Drop small spoonfuls of pesto over bowls of soup, garnish with slivered mint, and serve remaining pesto on the side.
Note: Nutritional analysis is per 1 1/2-cup serving.