I can see the fat under my skin in a cut. is this serious?

Full thickness: It depends on its size / location / depth / age + your general health (?Diabetic?Cancer?Hiv?Etc.). To see subcutaneous fat just means the layers of the skin have all been cut. These wounds are more prone to infection as often bleed less (from depth), so if it is concerning to you, should be seen at urgent care or dr's office (or er) for evaluation.