Prep the veg - Separate the cauliflower into bite-sized florets. Trim and peel the carrots, and slice them on the bias into 1/2-inch slices. Trim the celery, halve lengthwise if they stalks are very large, then slice them across into 1/2-inch slices. Destem and deseed the peppers. Slice the banana peppers. Chop the serrano pepper into a fine dice.
Make the pickling brine - Add the water and vinegar to a large saucepan. Stir in the salt and sugar, and warm over medium heat to dissolve, then bring the brine to a boil, and boil for 3 minutes. Remove the brine from the heat.
Steep the veg in the brine and prep your jars - Add the vegetables to the brine, cover, and steep for 1-2 minutes. Scoop the vegetables out of the brine into a large bowl. Add the bay leaves and peppercorns to the jars.
Fill the jars - Tightly pack the vegetables into the jars, the cover with the brine, leaving about 1/2-inch headspace. Spoon 1 T oil into each jar, cover, and refrigerate.
Notes
The Giardiniera will last in the fridge for up to 3 months, but I'm sure you won't let that happen!