Heat oil in a heavy pan over high heat. Add in broccoli and cook until broccoli begins to char on the bottom. Stir, season with salt and pepper and continue to cook until tender and charred on both sides. Remove to a bowl
Add in butter, onion and carrot. Cook, stirring frequently until tender. Lower heat and add in garlic, cook for an additional minute.
Add in chicken broth, broccoli, bay leaves, 1/2 & 1/2 and bring to a boil. Reduce and simmer until broccoli is completely tender, about 30 minutes.
Remove bay leaves and pour soup into a high speed blender, blend soup (only blend half of soup if you want a chunkier soup). Return soup to pan and add in cheese a cup at a time. Stir until cheese melts. Season with paprika and salt and pepper.