A roux is a mixture of equal parts of fat (butter, oil or meat rendering) and flour that you cook for two to four minutes over medium heat to remove the raw flavor of the flour and which thickens into the gravy when you add the liquid and heat. It is hard to make a lumpy gravy with a roux base.
To make a roux gravy, the basic formula is 1 tablespoons fat, 1 tablespoons flour, to 1 cup of liquid. For thicker gravy you can use 2 TBS flour- 2 Tbs fat and 1 1/2 cups of liquid. After cooking the meat or poultry remove the meat from the pan. Next you will need to ,find the amount of "juice" remaining in the pan drippings. I pour all the drippings into a glass measuring cup to see how much liquid there is and how much fat there is. Let it sit for a minute so that the fat rises to the top. Pour the fat off into another measuring cup and save to make your roux. If there is not enough liquid to make the amount of gravy you need then you can do one of two things. One add some boiling water to the bottom of the roasting pan and stir it around scraping all the browned bits off the bottom of the pan and use this liquid to add to the other juice from the meat or two- you can use canned broth. The amount of liquid you have will be the final amount of gravy so if you want one cup of gravy you will need one cup of liquid.
Now that you have the final amount of liquid you will know how much flour and fat you need. need to adjust the fat or oil amount up or down to suit your needs according to the basic formula. Lets say you have 2 cups of liquid. I would use 2 Tbs fat and 2 Tbs of white flour. Place the fat and the flour in a saucepan over medium heat. If you do not have enough fat from the meat then add butter to make the 2Tbs
Stir constantly over medium heat for 2 to 4 minutes until the flour just starts to get a slight golden color to it. This is a roux. I like using a whisk. Now add your liquid. Start by adding 1 3/4 cups into your roux. Stir constantly and kind of fast. If you don't stir you will get lumps. The mixture will start to thicken almost immediately. Continue to stir until the mixture comes to a simmer and bubbles. The gravy will thicken as it cooks and will also thicken more as it cools so resist the urge to add more flour. If it needs thinning you can add the rest of the broth.
Season to taste with salt and pepper, nothing is worse than under seasoned gravy, taste and serve.
You can also use a butter and flour roux with milk or cream as your liquid to make a white sauce- bechamel sauce and use it for pot pie or with sausage over biscuits.