In a large pot, melt the butter over medium heat.
Add the chopped onion and cook, stirring occasionally, until soft and translucent, about 5 minutes.
Add the minced garlic, smoked paprika, ground cumin, and cayenne pepper. Cook, stirring constantly, for 1 minute until fragrant.
Pour in the crushed tomatoes and vegetable broth. Stir 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. Cook over low heat, stirring frequently, until the cheese has melted and the soup is heated through, about 5 minutes.
Season with salt and freshly ground black pepper to taste.
Remove from heat and stir in the chopped fresh basil.
Ladle the soup into bowls and serve warm.