Dice the onion and carrot and peel and cube the potato.
Heat the oil in a large pot over medium heat and cook the onion and carrot, stirring, until soft, about 5 minutes.
Add the broth, potato and whole chicken breasts. Bring to a boil over high heat, then simmer gently over low heat for 15 minutes, until the chicken reaches 74°C (165°F). Lift out the chicken and shred it.
Add the broccoli and simmer over medium heat for 5 minutes until tender.
Blend about half the soup with a stick blender until smooth, leaving some chunks of broccoli.
Lower the heat to low and stir in the milk, then add the cheddar a handful at a time with the Dijon, stirring until melted. Do not let it boil or the cheese can turn grainy.
Return the chicken, season with the salt and pepper and serve.