Preheat a grill pan or panini press over medium heat for the panini. Preheat the grill or grill pan for the chicken.
Season the chicken breasts with salt and pepper. Grill the chicken an internal temperature of 160℉ (71℃), using a thermometer, about 4-5 minutes per side. Let it rest for a few minutes before slicing.
Assemble the panini: Spread the mayo on one side of each slice of bread and layer each with cheese, grilled chicken slices, and avocado slices. Top with the remaining slices of bread.
Lightly butter or brush olive oil on the outside of the panini and grill in the panini press or on the grill pan until the bread is toasted, and the cheese is melted.
Mix the lemon with the sparkling water.
Serve the panini with barbecue chips and pickle, and a glass of lemon sparkling water.