Cut steak into twelve 4x3-in. pieces; flatten to 1/4-in. thickness. Place a bacon piece and pickle along a short side of each; roll up and secure with toothpicks. In a skillet, heat 2 tablespoons oil over medium-high heat. Brown roll-ups on all sides. Remove and keep warm.
In the same skillet, heat the remaining oil. Stir in flour until smooth. Cook and stir over medium heat until golden brown, about 5 minutes. Gradually blend in broth and savory, stirring constantly. Return roll-ups to pan. Cover and cook over low heat for 1-1/2 hours or until tender. Season with salt and pepper.