Place 1 tablespoon of butter in a skillet over medium-high heat until it melts.
Add the onion and cook and stir it for about 5 minutes until it becomes tender and see-through. Set it aside.
In a large saucepan, mix ¼ cup of melted butter and flour together over medium-low heat. Keep whisking and cooking for 3 to 4 minutes. If needed, add 1 to 2 tablespoons of milk to prevent the flour from burning.
Gradually pour in milk while whisking all the time. Also, add chicken stock and bring it to a simmer. Cook for about 20 minutes until the mixture thickens.
Add broccoli, carrots, celery, and the sautéed onion. Simmer until all the vegetables become tender, which should take about 15 minutes.
Stir in Cheddar cheese until it melts. Season with salt and pepper according to your taste before serving.