Turkey bacon (check the ingredients for additives and sneaky sugar) or smoked fish, wilted greens (spinach, kale) and a poached egg – just because it's low carb/healthy doesn't mean it shouldn't feel fancy or indulgent for Sunday brunch. Top that egg with some nice maldon salt, or your favourite sugar free hot sauce and you're laughing.
Cheese, nuts, chia seeds, extra virgin olive oil, full fat yogurt for breakfast. You can put some olive oil on a Greek saladlike the Mediterranean do. Apart from these, fatty dish, dark chocolate, and coconut are options you can eat for other meals. You can use coconut oil when cooking an omelette for instance; this oil does not spoil when cooked.