But when I rolled over to his recipe for gravy and saw he had out the sieve, I went uh-oh. And sure enough, he had the drippings being skimmed of fat and then flour added to the liquid. That process almost always results in lumps.
For really good gravy that has a good color and no lumps, first make a roux. For each cup of liquid you have (I'm assuming you've simmered the neck and giblets while the chicken was cooking or have some good quality stock plus the defatted part of the drippings from the pan), put 1 tablespoon of the fat you skimmed off from the drippings into a large cast-iron skillet on medium heat until it's almost smoking. Add a like amount of flour and stir the mixture constantly until it's a deep brown, turning down the heat if it's browning too fast (if it burns, you will need to throw it out and start over). When it's a very deep brown, whisk in the liquid, turn up the heat a bit and bring it to a low boil. When it thickens, remove from the heat and serve. The only lumps would be the dripping solids and some bits of vegetable, and if you want it perfectly smooth, sieve your dripping liquid before adding to the pan.
I rarely use a sieve. Bring on the full taste!