Step 1

Heat 4 Tbsp. butter in a Dutch oven or medium stockpot over medium heat. Add onion and cook, stirring often, until softened and lightly browned around the edges, 8–10 minutes. Season with salt and pepper. Add tomato paste and cook, stirring to coat onion, until tomato paste is slightly darkened in color, about 2 minutes. Add canned tomatoes, vegetable broth, thyme, bay leaf, and sugar and bring to a simmer. Season with salt and pepper and cook until flavors are melded, 35–40 minutes. Remove from heat and stir in cream. Remove thyme stems and bay leaf and let cool slightly. Working in batches, purée in a blender until smooth.

Step 2

Heat a dry skillet, preferably cast-iron, over medium heat. Dividing evenly, spread room-temperature butter over bread slices. Top unbuttered side of 8 slices with cheddar, dividing evenly. Top with remaining bread, buttered side up, and cook, turning once, until bread is golden brown and cheese is melted, about 3 minutes per side. Cut sandwiches into soldiers and serve alongside hot soup for dipping.