Warm oil in a small saucepan over medium heat. Add mustard seeds and cook until they pop, about 2 minutes. All at once, add remaining spices except for turmeric and cook, stirring, until spices are very fragrant, about 1 minute. Add turmeric and stir just until sizzling. Carefully add 1/2 cup water, the vinegar, lime juice, sugar, salt, garlic, and ginger. Bring to a boil over high heat, then reduce heat and simmer for 2 minutes.
Pack cucumber spears into a deep bowl. Pour pickling liquid over them, weight with a small plate to keep submerged, and let cool to room temperature, about 40 minutes. Lift pickles out of brine to serve.
Make ahead: Chill, covered, up to 2 days (pickles will get more flavorful).
Note: Nutritional analysis is per spear.