Arrange a rack in center of oven; preheat to 375°. In a medium, high-sided, cast-iron or other ovenproof skillet over medium heat, heat 3 Tbsp. oil. Season chicken with salt and pepper and cook, turning occasionally, until golden brown and an instant-read thermometer inserted into the center registers 165°, 5 to 7 minutes per side. Transfer to a plate and let cool slightly.

In same skillet over medium-low heat, cook tomatoes, stirring occasionally, until slightly darkened and brick red, about 2 minutes. Stir in garlic and thyme and cook, stirring, until fragrant, about 1 minute more. Add a splash of water, scraping up any browned bits from bottom of skillet with a wooden spoon. Remove from heat.

In a food processor, pulse cream cheese, about three-quarters of tomato mixture, any bits from bottom of skillet, half of Parmesan, and 1/4 tsp. salt until smooth. Transfer to a large bowl.

Using your hands or 2 forks, shred chicken. Fold into cream cheese mixture. Spoon mixture back into skillet and spread in an even layer.

In a small bowl, combine cheddar and remaining Parmesan and tomato mixture. Sprinkle over top. Place skillet on a baking sheet.

Bake dip until top is golden and bubbly, about 20 minutes.