Separate chard stems and leaves. Fold each leaf in half and cut out the stem. Thoroughly wash stems, then dice. Roll leaves into a big bundle, then finely slice. Transfer leaves to a salad spinner and submerge leaves in water, then lift out the strainer basket to rinse. Repeat several times. Spin leaves dry and set aside.
Melt butter over medium heat in a stockpot. Add onions and chard stems, sauté until soft about 5 minutes. Sprinkle flour in and stir until a smooth paste is formed. Add stock 1 cup at a time, whisking until smooth. Stir in milk and bring to a simmer. Add chard leaves in batches (about 4 – 6 cups leaves total, depending on desired consistency) and simmer soup about 5 – 8 minutes more until chard is softened, but still green.
Purée soup, using an immersion blender or blender. Use caution when blending hot liquids! Stir cheese into puréed soup and serve warm, garnished with more cheese.
This soup is great on it’s own, but I enjoyed the added crunch of fresh croutons.