In a large stock pot, heat oil over medium heat. Add celery, carrots, and onions. Saute until soft - approximately 8-10 minutes. Add garlic, salt, pepper and cook for one minute or until fragrant.
Add broth, water, and split peas. Bring to a boil.
Reduce heat to a simmer. Cover. Cook for 50-55 minutes or until peas are falling apart and tender.
Add spinach. Cook another 5 minutes. Remove from heat.
Working in batches, puree in the blender. Add pesto to one of the batches before blending. Stir to thoroughly combine into all the soup.