Preheat oven to 350°. Heat 3 Tbsp. vegetable oil in a small skillet over low until shimmering. Add 1 garlic clove, finely chopped, and 1 tsp. crushed red pepper flakes and cook, stirring constantly, until fragrant but not browned, about 2 minutes. Pour garlic-chile oil into a medium bowl.
Add 4 oz. cream cheese, room temperature, 4 oz. sour cream, 1 cup dill relish, 1 oz. Parmesan, finely grated, ¼ cup chopped dill, 1 Tbsp. finely grated lemon zest, 1 Tbsp. fresh lemon juice, and 1 tsp. freshly ground black pepper to garlic-chile oil and whisk to combine; season dip with kosher salt. Scrape into a 1-qt. baking dish and smooth surface.
Place 1½ cups Cheez-It crackers in a resealable plastic bag, seal, and smash with a large wooden spoon, rolling pin, or frying pan into coarse crumbs. Scatter over dip and bake until warmed through, about 10 minutes.
Serve dip warm with crudités and more crackers for dipping.
