Here's the page on Thyroid disorders that @Carorees wrote up.
https://www.fastday.com/fasting/intermi ... disorders/Yes, the thyroid very much follows a circadian rhythm. We have the lowest amount of T3 and highest amount of TSH in the morning, before we eat. If you are taking hormone replacement, this is why you take it first thing in the morning on an empty stomach. One good friend of mine found that simply making sure she kept her medication by her bed, taking it first thing on an empty stomach and waiting at least 1/2 hour to eat or drink anything other than water was all she needed to do to feel better and lose weight normally. Before doing this (she had been taking her medication later in the morning, with breakfast), she had no energy, was gaining weight way too easily and felt terrible.
From all of the things I've read on hypothyroid, the 800 calories/calorie composition in this book must be a guess. I haven't seen any studies about specific diets or specific calorie amounts targeting specifically hypothyroid people. It may be a good guess, but it's most likely a guess.
Most people looking to lose weight will do better by eating fewer carbs. Since too much protein is also a problem, it naturally follows that some increase in fat consumed is necessary, and healthier fat is always a better choice than hydrogenated oil. I don't generally go low-carb, but I do go low carb on fast days (although I'll make an exception for beans in a soup, like yummy black beans).