The Buff-Tip Moth: the resting posture, shape, and color/pattern of the buff-tip moth allows it to mimic a broken birch twig; the moth's buff-colored head and the patches on its hindwings even resemble freshly-snapped wood
It also definitely tends to look like there's a weird little smiley face in the setae surrounding the moth's head, which is arguably even more striking but for some reason none of the sources I dug up on this species seem to mention that weirdness.
This type of camouflage is generally referred to as a "protective resemblance" -- a form of mimesis in which an animal can avoid being preyed upon by mimicking an inedible/unremarkable aspect of its environment. Many different moths are able to disguise themselves in similar ways, and it is especially common within the family known as Lasiocampidae. Some of the other species that engage in protective resemblance include Gastropacha quercifolia, Gastropacha pardale, Gastropacha populifoli, Euthrix potatori, Euthrix laeta, and Calyptra minuticornis (along with the other members of genus Calyptra). Most of these moths disguise themselves as leaves/foliage.
The buff-tip moth is particularly adept at disguising itself, however, and the fact that it so strongly resembles such a specific object (i.e. not just a dead leaf or a vague piece of foliage -- but a broken twig from a silver birch tree, in particular) makes this disguise seem even more impressive/unique.
This species (Phalera bucephala) can be found throughout the British Isles, mainland Europe, and Asia, with its range extending into Eastern Siberia.
Sources & More Info:
Wildlife Insights: Buff-Tip Moth Identification Guide
ButterflyConservation.org: Buff-Tip Moth
The Wildlife Trusts: Buff-Tip Moth
Wildlife Insight: the Buff-Tip Moth
Moth Identification: P. busephala
Encyclopedia of Life: Global Map of Known Occurrences for P. busephala
Insecta: Phalera bucephala
Lepidoptera and their Ecology: P. busephaloides and P. busephala
Journal of Ecology & Evolution: Strong Foraging Preferences for Ribes alpinum in the Polyphagous Caterpillars of Buff-Tip Moth Phalera bucephala
Dickinson County Conservation Board: Protective Resemblance & other Forms of Mimesis/Mimicry