Bring a 6-quart saucepan of water to a boil. Place 6 one-pint canning jars along with their bands and lids in the boiling water. After about 30 seconds, remove the lids from the water with tongs and transfer to a kitchen towel on the counter. After about 10 minutes, use tongs to transfer the jars and bands to the kitchen towel and let air-dry.
When the jars are dry, add 1/2 tsp. coriander seeds, 1/4 tsp. turmeric, 1/4 tsp. cumin seeds, 1/4 tsp. black mustard seeds, 1 bay leaf, and 1 dried, split chile de arbol to each jar.
Bring a 4-quart saucepan of salted water to a boil. Cut cauliflower into 1" florets. Peel and slice the carrot into 1/8"-thick coins. Halve and thinly slice the onion.
Boil the cauliflower and carrots for 3 minutes; drain and transfer to a large bowl.
Add the onion and stir to combine. Divide the vegetables between the jars, leaving about 3/4" of space at the top of each jar.
In a 4-quart saucepan, bring 4 cups water, sugar, and salt to a boil over high heat, whisking occasionally to dissolve it, about 5 minutes.
Pour the vinegar solution into the jars, leaving 1/2" of space below each rim. Wipe the rims of the jars with a hot damp towel. Cover and seal each jar with a lid and screw the bands on tightly.
Place the jars in a large pot fitted with a rack; pour in enough water to cover the jars by at least 3". Bring to a boil over high heat; boil for 15 minutes. When using this recipe at altitudes of 1,001 to 6,000 feet, add 20 more minutes of processing time; above 6,000 feet, add an additional 5 minutes. Turn off the heat; let sit for 5 minutes. Using tongs or a rubber-handled jar lifter, transfer the jars to a kitchen towel on the counter; let cool for 6 hours.
Store cauliflower in a cool, dark place for up to 1 year. Refrigerate after opening.