In a large pot, melt the butter over medium heat.
Add the chopped onion and cook, stirring occasionally, until softened and translucent, about 5 minutes.
Add the minced garlic, smoked paprika, ground cumin, and cayenne pepper. Cook for 1 minute until fragrant.
Stir in the tomato paste and cook for 2 minutes to deepen the flavor.
Add the crushed tomatoes, vegetable broth, sugar, salt, and black pepper. Stir to combine.
Bring the soup to a simmer, then reduce the heat to low. Cover and let simmer gently for 15 minutes.
Remove the pot from heat and use an immersion blender to puree the soup until smooth. Alternatively, carefully transfer the soup to a blender in batches and blend until smooth, then return to the pot.
Place the pot back on low heat. Stir in the half-and-half and shredded pepper jack cheese until the cheese is fully melted and the soup is creamy.
Heat gently for 3-5 minutes, stirring frequently, but do not let it boil.
Taste and adjust seasoning with additional salt and pepper if needed.
Remove from heat and stir in the chopped fresh basil.
Serve hot, garnished with additional shredded pepper jack cheese or basil if desired.