Preheat oven to 350°F with 1 oven rack in middle position and a second rack about 6 inches from heat source.
Heat sun-dried tomato oil in a 10-inch cast-iron skillet over medium. Add sun-dried tomatoes; cook, stirring often, until softened, about 1 minute. Add garlic, thyme, crushed red pepper, pepper, and salt; cook, stirring constantly, until garlic is softened and fragrant, about 30 seconds. Remove from heat and let cool slightly, about 2 minutes.
Stir cream cheese, Boursin, and half and half into sun-dried tomato mixture until well combined and smooth. Stir in 1/4 cup Parmesan. Fold in chicken, 1/2 cup mozzarella, and 1/2 cup cheddar until evenly combined. Spread dip into an even layer. Sprinkle evenly with remaining 1/2 cup mozzarella, 1/2 cup cheddar, and 1/4 cup Parmesan.
Bake on middle rack until cheese is melted and edges are bubbling, 18 to 20 minutes. Increase oven temperature to broil. Transfer skillet to upper rack; broil until cheese is golden brown in spots, about 1 minute. Garnish with basil, and serve warm with toasted baguette slices.