Rinse the split peas in cold water; pick them over and discard any damaged or malformed peas or small stones. Set aside.

Dice the bacon.

Add the diced bacon to a Dutch oven or stockpot and place it over medium heat. Cook the bacon until it is crisp, stirring frequently. With a slotted spoon, remove the bacon to paper towels to drain. Refrigerate the bacon and save it to use for garnishing the soup.

In the bacon drippings over medium heat, cook the onion and celery for about 10 to 12 minutes, or until the onion is translucent and browned.

Add the split peas, water, broth, potatoes, carrots, pepper, and bay leaf to the Durch oven. Bring the soup to a boil over high heat and then reduce the heat to low. Cover the pan and simmer the soup for 45 minutes, or until peas are soft. Stir occasionally.

Discard the bay leaf. Carefully purée the hot pea soup in batches, filling the blender no more than 1/3 to 1/2 full each time. Make sure the lid is secure when blending the hot mixture, and hold the lid down firmly with a folded kitchen towel in hand. Pour the blended mixture into a bowl while blending subsequent batches.

Return the pureed soup to the Dutch oven; add the half-and-half and stir to blend. Cook over medium heat until hot, but do not boil. Taste and adjust the seasonings with salt and pepper, as needed.

Sprinkle the reserved bacon over servings just before serving.