Combine broccoli and broth in a medium-sized pot, then cover and simmer for approximately 10 minutes until the broccoli is tender. Separate the chicken broth and set it aside for later use, keeping the broccoli aside as well.

In a large pot over medium heat, melt butter. Add onions and cook until they become translucent. Incorporate flour and whisk continuously for around 2 minutes until the flour achieves a golden hue.

While stirring, gradually pour in the reserved chicken broth and milk. Simmer the mixture, stirring occasionally, for about 5 minutes until the broth thickens.

Integrate cheese, ham, broccoli, pepper, and salt to taste generously. Heat the soup until the cheese is melted and the entire mixture is warmed through.