Preheat oven to 375 degrees.
In a medium, high-sided cast iron or other ovenproof skillet over medium heat, heat 3 Tbsp. olive oil. Season chicken with salt and pepper and cook until golden brown. Instant read thermometer should read 165 degrees; 5 to 7 minutes if full-sized breasts; 3-5 minutes if skinny breasts. Transfer to a plate and let cool.
In the same skillet over medium-low heat, cook sun-dried tomatoes, stirring occasionally until darkened and brick red in color; about 2 minutes. Stir in garlic and thyme and cook, stirring until fragrant; about 1 more minute. Add a splash of water, scraping up any browned bits from the bottom of the skillet. Remove from the heat.
In a food processor, pulse cream cheese, about 3/4 of the tomato mixture and any brown bits from the bottom of the skillet. Add 1 cup Parmesan, 1/4 cup Gruyere and 1 cup cheddar. Also add 1/4 tsp. salt. Blend until smooth. Transfer to a large bowl.
Using your hands or 2 forks, shred the chicken. Fold into the cream cheese mixture. Spoon the mixture back into the skillet and spread in an even layer.
In a small bowl, combine the remaining cheddar, Parmesan and Gruyere. In layers, with cheeses being first, top the dip with cheese, remaining tomato mixture and finally top with the cheese. Place in the oven and bake until the top is golden and bubbly; about 20 minutes.