Stormdancer / Jay Kristoff.

Enhanced descriptions from Syndetics:

The first in an epic new fantasy series, introducing an unforgettable new heroine and a stunningly original dystopian steampunk world with a flavor of feudal Japan.

A DYING LAND
The Shima Imperium verges on the brink of environmental collapse; an island nation once rich in tradition and myth, now decimated by clockwork industrialization and the machine-worshipers of the Lotus Guild. The skies are red as blood, the land is choked with toxic pollution, and the great spirit animals that once roamed its wilds have departed forever.

AN IMPOSSIBLE QUEST
The hunters of Shima's imperial court are charged by their Shogun to capture a thunder tiger - a legendary creature, half-eagle, half-tiger. But any fool knows the beasts have been extinct for more than a century, and the price of failing the Shogun is death.

A HIDDEN GIFT
Yukiko is a child of the Fox clan, possessed of a talent that if discovered, would see her executed by the Lotus Guild. Accompanying her father on the Shogun's hunt, she finds herself stranded: a young woman alone in Shima's last wilderness, with only a furious, crippled thunder tiger for company. Even though she can hear his thoughts, even though she saved his life, all she knows for certain is he'd rather see her dead than help her.

But together, the pair will form an indomitable friendship, and rise to challenge the might of an empire.