Melt butter in a saucepan over medium heat; cook and stir onion and thyme in the melted butter until onion is slightly softened, about 5 minutes. Add vermouth; cook and and stir until liquid is evaporated, 2 to 3 minutes.
Pour liquid from artichoke hearts into a measuring cup; add enough chicken broth to equal 1 1/2 cup.
Transfer onion mixture to a blender and top with drained artichoke hearts; blend, adding chicken broth mixture as needed, until mixture is smooth. Pour artichoke mixture back into the pan and mix in remaining chicken broth mixture and blue cheese dressing. Bring to a simmer and cook, stirring often, until bisque is heated through, about 5 minutes. Serve warm or cold.