Heat a large skillet or wide saucepan over medium until hot, about 3 minutes. Add oil, and swirl to coat skillet. Arrange sausage patties in skillet in an even layer. Increase heat to medium-high, and cook, breaking up sausage with a wooden spoon and scraping bottom of skillet to release browned bits, until meat is browned and no pink remains, 4 to 5 minutes. (If necessary, reduce heat to medium or medium-low to prevent bitter-tasting burned bits.)
Transfer cooked sausage to a heatproof bowl, and set aside; discard any sausage drippings in skillet. (Do not wipe skillet clean.) Add butter to skillet, and melt over medium. Whisk in flour, and cook, whisking constantly, until mixture darkens to the color of peanut butter, 6 to 8 minutes.
Stir chicken broth, milk, thyme, black pepper, salt, garlic, cayenne, and cooked sausage into roux in skillet. Bring to a simmer over medium, stirring occasionally. Simmer, stirring often, until gravy is thickened, 8 to 10 minutes. Serve immediately with biscuits.
