Mix 2 tbsp. boiling water, the garlic, and chile; let steep 3 minutes. Stir in remaining chimichurri ingredients; season with salt.
Halve rolls and toast, cut side up, in a 350° oven until crisp, about 10 minutes.
Heat oil in a large heavy frying pan over medium heat and add sausages. Cook, covered, turning often, until cooked through, 15 to 20 minutes. Top each with cheese, remove from heat, and cover until cheese melts. Sprinkle with paprika.
Spoon 2 to 3 tbsp. chimichurri onto roll halves and make a sandwich with sausages. Serve with remaining chimichurri.