It is not safe to cook the turkey if it has been left out at room temperature for an extended period and is still partially frozen inside. Leaving meat, especially poultry like turkey, at room temperature allows bacteria to multiply rapidly and can lead to foodborne illnesses.
When thawing a turkey, the best and safest method is to do so in the refrigerator. It takes longer, but it keeps the turkey at a safe temperature throughout the thawing process. For an 11 lb turkey, it usually takes about 24 hours per 4 to 5 pounds of turkey weight to thaw in the refrigerator. So, an 11 lb turkey should ideally be placed in the refrigerator 2-3 days before you plan to cook it.
Since you mentioned that the turkey is still cold on the outside, it might be salvageable if you immediately transfer it to the refrigerator and allow it to thaw properly until it's completely thawed throughout. However, you should closely monitor the turkey's internal temperature and make sure it stays below 40°F (4°C) to prevent bacterial growth.
If, by tomorrow, the turkey is still partially frozen, or if you're unsure about its safety, it's better to err on the side of caution and not cook it. Consuming improperly thawed or spoiled meat can lead to food poisoning and other health risks.
In the future, always plan ahead and allow enough time for proper thawing in the refrigerator to ensure a safe and delicious turkey for your meals. If you need a quicker thawing method, you can also use the cold water method by placing the turkey in a leak-proof plastic bag and submerging it in cold water. Change the water every 30 minutes to keep it cold. However, cooking the turkey immediately after thawing using this method is essential to prevent any bacterial growth.