Melt butter with olive oil in a Dutch oven or large pot over medium heat.
Add onions, carrots, celery, Italian seasoning, cayenne pepper, salt, and black pepper. Cook, stirring occasionally, for 7-10 minutes, or until vegetables are tender.
Add garlic and cook until fragrant, about 1 minute.
Add in the chicken broth, Dijon mustard, chicken, ham and bay leaf. Bring to a simmer.
Cook at a simmer for about 10 minutes, stirring frequently. Add shredded Swiss cheese and stir until incorporated and cheese is melted.
Stir in the heavy cream and half and half and continue cooking until heated through. Test for seasoning.
When you’re ready to serve, discard the bay leaf. Spoon into bowls and garnish with shredded Swiss cheese, croutons and Italian seasoning. Enjoy.
