Heat oil in a skillet over medium-high heat; saute onion until translucent, about 5 minutes. Set aside.

Melt butter in a soup pot over medium heat and add flour. Cook until roux is bubbly and beginning to turn brown, 3 to 4 minutes. Pour milk in gradually while whisking constantly. Add water and vegetable bouillon cube. Add carrots, celery, cooked onion, potatoes, kale, black pepper, and cayenne. Simmer, stirring occasionally, until vegetables are tender, about 45 minutes.

Add shredded Cheddar cheese to soup and stir until melted.