Love this recipe (so tasty!), but the sauce keeps breaking (all the fat separates out). What am I doing wrong?

The ratio of flour to fat might be off, you need enough flour to absorb the fat or your gravy will break. Chicken thighs can vary as far as the amount of fat is concerned and it's possible the thighs you used were more fatty than those used in the original recipe. You can add a little more flour maybe 2 tsp to the cream and see if that helps. When I make gravy if I don't have the right ratio of flour to fat it will break. I have to make this, had forgotten about it and it sounds delicious.