Heat oven to 425°F. Toss carrots with olive oil and salt on a sheet pan and roast for 25 to 30 minutes, turning once, until tender and browned at the edges.
Melt butter in a soup pot over medium heat. Cook leeks for 5 minutes until softened, then stir in garlic and cumin for 1 minute.
Add split peas, broth and pepper. Bring to a simmer, partly cover and cook over low heat for 45 to 55 minutes, stirring from the bottom occasionally, until the peas collapse easily.
Add roasted carrots and simmer for 5 minutes. Remove from heat and blend carefully with an immersion blender until smooth, keeping the blade submerged.
Fold in ham and return to low heat for 5 minutes until the meat reaches 165°F. Stir in lemon juice and add up to the measured cup of water if needed for a thick but pourable soup.
Divide among six bowls, making sure each portion includes pieces of ham.