Melt butter with olive oil in a large stockpot over medium heat. Cook and stir onion and garlic in hot butter-oil until onion is slightly softened, about 5 minutes. Add flour, dill seed, tarragon leaves, and black pepper to onion mixture; cook and stir until flour is dissolved and fragrant, about 3 minutes.

Pour chicken broth over onion mixture and stir until blended, 2 to 3 minutes. Add cabbage and corned beef; cook, stirring frequently, until cabbage is tender, about 20 minutes. Stir in half-and-half and continue to cook just until soup is warm but not boiling, about 5 minutes.

Meanwhile, melt cream cheese in a small saucepan over low heat until creamy, 2 to 3 minutes. Whisk together cornstarch and water in a small bowl until smooth; stir into melted cream cheese until well combined.

Stir cream cheese mixture into soup; cook over low heat, stirring constantly, until cheese melts and soup is thickened, about 5 minutes.