In a medium-size pan, saute onion in olive oil on low heat until it is just starting to brown. While onions cook, cut asparagus into 2-inch pieces, reserving tips.
Add asparagus stalks to pan with stock. Bring to a boil and simmer for approximately 10 minutes, until just tender. Time will vary depending on thickness of asparagus stalks.
Add cannellini beans and blend soup until creamy. A traditional or immersion blender will make it easy.
When ready to serve, heat soup to a simmer and cook asparagus tips in soup for about 2 minutes, just until barely tender. Lemony Pesto mixture can be swirled into individual bowls or combined with soup right before serving.
Grind all ingredients with a mortar and pestle or process briefly in a food processor.