Preheat oven to 300°F. Coat a 9-by-13-inch baking dish with cooking spray. Spread broccoli evenly in the dish.
Tear bread into pieces and process in a food processor until coarse crumbs form. Spread the breadcrumbs on a large rimmed baking sheet and bake until dry and crispy, 8 to 10 minutes. Remove from oven and increase oven temperature to 350°F.
Heat 2 tablespoons sun-dried tomato oil in a large saucepan over medium heat. Add onion and garlic; cook, stirring, until soft, 3 to 5 minutes.
Sprinkle flour over the vegetables and cook, stirring, for 1 minute. While stirring, slowly pour in broth. Cook, stirring occasionally, until thickened, 3 to 5 minutes.
Stir in cream cheese, sour cream, salt and pepper; cook, stirring, until smooth, about 2 minutes.
Remove from heat and stir in 1 cup Cheddar and sun-dried tomatoes.
Pour the sauce over the broccoli in the baking dish and gently stir to distribute evenly. Sprinkle the remaining ½ cup Cheddar over the top.
Bake until the sauce is bubbling around the edges, 20 to 25 minutes.
Meanwhile, drizzle the remaining 1 tablespoon oil over the breadcrumbs and toss to coat. Scatter over the casserole when it's done baking.
