Rinse and drain the yellow split peas. Bring 4 cups of water to a boil in a medium-sized sauce pan. Add the yellow split peas and simmer until the split peas are soft. About 30 minutes. You might need to add more water.
In a large stockpot, heat up the olive oil until shimmering. Add the diced onion and, stirring occasionally, sauté on medium-low heat until browned. About 10-15 minutes, being careful not to burn.
Add the garlic and sauté for an additional 1-2 minutes. Stir in the tomato paste and increase the heat to medium. Allow the tomato paste to caramelize until it is a dark red.
Add the celery and potatoes and stir to coat. Add salt and pepper to season. Start with 1 tsp of salt and a few grinds of fresh pepper. You will likely want to add more as you taste and cook.
Add the vegetable broth. Simmer until the potatoes are able to be easily pierced with a fork.
In the final 10 minutes of simmering the yellow split peas, add the carrots to the pot with the yellow split peas.
When the split peas and carrots are soft, there should only be a small amount of cooking water remaining. I like to add that to the vegetable broth when I add the carrots and split peas. It usually amounts to ~1 cup of cooking water. If you have more than that, you can drain the excess water before adding.
Add the fresh lemon juice and any additional salt and pepper.
Blend the entire pot with a few pulses of an immersion blender. I like to leave some larger pieces in the soup.
Add the baby spinach and stir until slightly wilted. Serve immediately.