Heat vegetable oil in a large pot over medium-high heat. Add sliced kielbasa and cook until browned on both sides, approximately 7-9 minutes. Set aside on a plate to keep warm.
In the same pot, melt butter. Add onion, carrots, minced garlic, salt, and pepper. Stir to combine and cook for an additional 10-12 minutes, until the onion becomes translucent and the carrots begin to soften.
Sprinkle flour over the vegetables, stirring well. Continue cooking for another minute.
Pour in chicken broth, whole milk, and Worcestershire sauce. Stir to combine and bring the mixture to a simmer.
Return the kielbasa to the pot and add the beans.
Reduce heat to medium-low and simmer, stirring occasionally, until the mixture thickens, approximately 15-20 minutes.
Garnish with parsley before serving.
Enjoy this delicious and hearty Kielbasa and White Bean Soup!