For the honey giardiniera butter: Add the butter, honey and salt to taste to a bowl and mix until smooth. Stir in the giardiniera until well blended. Refrigerate until ready to use.
For the chicken: Add the chicken thighs, reserved giardiniera brine and pickle brine to a plastic zip-top bag and refrigerate overnight or for at least 2 hours.
When ready to fry the chicken, fill a Dutch oven halfway full with oil and heat to 350°F.
Set up a dredging station: Pour the buttermilk into a large bowl and in a separate large bowl, stir together the flour, cornstarch, paprika, salt, pepper and cayenne until well combined. Use your tongs to drizzle some of the buttermilk mixture into the flour mixture.
Remove 1 or 2 pieces of the chicken from the brine at a time and submerge in the buttermilk. Add 1 piece of the buttermilk-soaked chicken at a time into the flour mixture and turn to completely coat the chicken. Transfer to a plate and repeat until all the remaining chicken is well coated.
Add the chicken to the hot oil in batches, 2 or 3 pieces at a time, and cook until brown and crispy, 12 to 15 minutes. Transfer the cooked chicken to a wire rack on a baking sheet to drain. Serve immediately with the giardiniera butter or let cool and store on a wire-rack on a sheet pan, lightly covered, in the fridge overnight. Serve at the picnic at room temp. Served with softened honey giardiniera butter.