Heat a large soup pot over medium-low heat; add olive oil and heat until it starts to smoke. Decrease heat to low and add onion, bacon, celery, garlic, and rosemary to the hot oil. Cover pot with a lid and cook until onion is tender and bacon is cooked through, 3 to 5 minutes.

Mix squash, carrot, paprika, and cayenne pepper into onion-bacon mixture; return lid and increase heat to medium-low. Cook squash mixture, stirring occasionally, until squash is softened and nearly cooked through, about 20 minutes.

Pour chicken stock over squash mixture and increase heat to medium-high; cook until squash is cooked through, about 10 minutes more. Remove pot from heat and cool soup slightly, 2 to 3 minutes.

Blend soup using an immersion blender until smooth; season with salt and pepper. Add more chicken stock for a thinner soup.