Melt the butter in a large pot over medium-high heat. Add the onion, ½ teaspoon salt, and ¼ teaspoon pepper. Cook, stirring occasionally, until tender, 4 to 6 minutes. Add the broth, peas, and romaine, cover, and bring to a boil, stirring occasionally. Add the tarragon or mint and remove pot from heat. (The peas and herbs should be bright green.) Puree the soup, in batches if necessary, until smooth. (A regular or immersion blender will both work.)
Meanwhile, heat broiler. Top the ciabatta rolls with the ham and Cheddar, dividing evenly. Broil on a foil-lined baking sheet until the cheese is melted, 1 to 2 minutes.
Serve the soup with the cheese tartines on the side.
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