Was not easy.
If I'd known at the time I took this picture what I know now, I might not have tried for it. It turns out that when you get to close to a thrasher nest, you can actually get attacked by the bird. Cornell has records of Thrashers actively attacking humans and even drawing blood.
In addition to their distinctive color and shape, thrashers can be noticed by their call-- a wide range of songs that tend to be repeated in pairs. Catbirds will go through songs one after the other, and Northern Mockingbirds tend to do repeats in 3+ iterations. Whenever you hear a bird in the US with exactly two repetitions of a large variety of songs, you're probably dealing with a brown thrasher.