Preheat the oven to 350 degrees F (175 degrees C). Spray a baking pan with cooking spray.

Heat a skillet over medium heat. Add oil, garlic, and onion. Cook and stir until tender, 1 to 3 minutes. Add ham and spinach leaves and stir until wilted, about 1 minute. Chop slightly in the skillet. Remove from heat and stir in cream cheese until melted and mixture has cooled a bit. Add Swiss cheese and set filling aside.

Cut each chicken breast to 1/2 inch thickness and place in a bowl. Pour Italian dressing over chicken. Place 1 chicken breast on a cutting board and pound thinner. Add about 1/4 of the filling and fold chicken over to form a packet. Seal using toothpicks and place in the prepared baking pan. Repeat with remaining chicken breasts and filling.

Combine crushed crackers and Cheddar cheese together in a bowl. Cover filled chicken breasts with cracker mixture, mashing them into the breasts.

Bake in the preheated oven for 45 minutes. Remove from oven and let sit for 10 minutes before transferring from the pan to a serving platter to allow filling to set up. Remove toothpicks before serving.