Bring chicken broth to a boil in a large pot. Cook cauliflower, white wine, and bay leaves in the boiling broth over medium-high heat until cauliflower is tender, 10 to 15 minutes.
Mix garlic, cream cheese, and Asiago cheese together in a bowl.
Strain cauliflower from broth with a slotted spoon and add to the cheese mixture. Remove bay leaves from stock and discard.
Mash cauliflower and cream cheese mixture with a potato masher until well mixed; season with salt and black pepper.
Stir cauliflower mixture into the broth until smooth. Season with rosemary, thyme, salt, and black pepper.
