Melt butter and olive oil together in a stockpot over medium heat; saute onion and garlic 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. Add half-and-half and cook just until liquid is warm but not boiling, about 5 minutes.

Stir half-and-half into the pot; continue to cook just until liquid is warm but not boiling, about 5 minutes.

Melt cream cheese over low heat in a saucepan until creamy, 2 to 3 minutes. Whisk cornstarch and water together in a bowl; stir into cream cheese until smooth. Stir cream cheese mixture into soup; cook over low heat, stirring constantly, until cheese melts and soup is thickened, about 5 minutes.