Heat oil and add vegetables: In a Dutch oven or 5-quart heavy pot with a lid, heat oil over medium. Add onion, carrots, celery, and thyme; season with salt and pepper.
Sauté vegetables: Cook until vegetables begin to soften, 5 to 8 minutes.
Simmer split peas and ham bone in broth: Add broth, split peas, ham bone, and 5 cups water. Bring to a boil, reduce heat to medium-low, and partially cover; simmer until peas are soft, 30 to 45 minutes.
Toast croutons: While soup is cooking, make croutons: In a large skillet, melt butter over medium heat. Add bread and cook, tossing occasionally, until browned and crisp, 6 to 8 minutes.
Move croutons to plate: Transfer to a paper-towel-lined plate.
Puree soup: Remove and discard bone from soup. Working in batches, puree only ½ the soup in a blender (don't overfill); return to pot.
Heat: Add ham cubes, and simmer until heated through. If necessary, thin with water.
Serve with ham and croutons: Add salt, pepper, and lemon juice to taste. Serve topped with croutons
