In a large pot, melt the butter over medium heat.
Add the chopped onion and sauté until translucent and soft, about 5 minutes.
Stir in the minced garlic, smoked paprika, ground cumin, and cayenne pepper. Cook for 1 minute until fragrant.
Pour in the crushed tomatoes and vegetable broth, stirring to combine.
Bring the mixture to a simmer and cook uncovered for 15 minutes, stirring occasionally.
Using an immersion blender, carefully blend the soup until smooth. Alternatively, transfer the soup in batches to a blender and puree until smooth, then return to the pot.
Stir in the whole milk and shredded pepper jack cheese until the cheese is fully melted and the soup is creamy.
Add sugar, salt, and freshly ground black pepper to taste. Stir well.
Simmer the soup for an additional 5 minutes, stirring occasionally.
Remove from heat and stir in the chopped fresh basil.
Ladle the soup into bowls and serve warm.