Heat a large pot over medium heat. If using the bacon, add the bacon and cook, stirring occasionally, until crisp and the fat is rendered, 8 to 9 minutes. Reduce the heat to medium-low if necessary to avoid browning the bacon too quickly. With a slotted spoon, transfer the bacon to a paper towel-lined plate. Pour off all but 1 tablespoon of the drippings.

Return the pot to medium heat. If you did not cook the bacon, add the oil. When hot, add the sausage. Cook, stirring occasionally and breaking the meat into smaller pieces with a wooden spoon, until crisp and browned all over, about 5 minutes (it’s ok if there are still some pink spots). With a slotted spoon, transfer the sausage to a paper towel-lined plate, leaving the drippings in the pot (it can be the same plate as the bacon, but keep the two separated).

Return the pot to medium heat and add the onion. Cook, stirring, until softened and translucent, about 5 minutes. Add the garlic, salt, and pepper and cook for 1 minute, until fragrant.

Add the broth and potatoes. Increase the heat to high and bring to a rapid simmer. Reduce the heat to low to maintain a simmer. Partially cover the pot and cook for 6 to 8 minutes, until the potatoes are fork-tender (if you cut them larger than 1/4-inch, you’ll need longer).

Turn off the heat and use an immersion blender to partially puree the soup. Or transfer about 2 cups of the soup to a high-speed blender. Put the lid on the blender, remove the center piece of the lid to let out steam, and cover the hole with a kitchen towel. Puree until smooth, then return to the pot.

Bring the soup to a simmer and add the sausage and kale. Cook, stirring to submerge the kale, until it’s just wilted, 2 to 3 minutes. Stir in the half and half and lemon juice and season to taste with additional salt and pepper.