Step 1Combine the wild rice and 2 cups of the chicken broth in a large saucepan. Bring to a boil. Cover and boil gently 30 minutes. Turn off the heat and let stand on the burner, covered, for 30 minutes.
Step 2Heat the butter in a 3-quart saucepan over low heat until melted. Add the onion and cook until tender, but not browned, about 5 minutes.
Step 3Trim the dried ends from the broccoli stems. Chop the broccoli and stems into 1/2-inch pieces. Add them to the pan with the onion along with 2 cups of chicken broth. Bring to a boil. Reduce the heat, cover and simmer until the broccoli is tender, about 5 minutes.
Step 4Stir in the half-and-half. Bring to a simmer. Blend together the flour and the remaining 1/2 cup of chicken broth until smooth, then whisk into the hot soup. Bring the soup to a boil, stirring constantly, and continue boiling until it thickens, about 1 minute. Drain the cooked rice and stir into the soup along with the lemon juice and minced sage. Add salt to taste.