Season the chicken all over with salt and pepper. Mix the flour with 2 teaspoons rosemary and 1 1/2 teaspoons pepper in a shallow bowl. Put the buttermilk in another shallow bowl. Spoon 2 tablespoons buttermilk into the flour mixture and stir to make clumps. Dip the cutlets one at a time in the flour, then in the buttermilk, then back in the flour again to make a shaggy coating. (You should use all of the flour mixture.) Set the chicken aside on a plate.
Heat 2 tablespoons olive oil, the garlic and remaining 1 teaspoon rosemary in a medium saucepan over medium-high heat. Cook until the garlic is lightly toasted, about 1 minute. Stir in the cannellini beans and their liquid, 3/4 cup water and a pinch of salt. Adjust the heat to maintain a steady simmer and cook until slightly thickened, 10 to 12 minutes; add a splash of water if the mixture is too thick. Season with salt and pepper. Remove from the heat and cover to keep warm.
Meanwhile, heat 1/4 inch olive oil in a large skillet over medium-high heat. Working in batches if necessary, add the chicken and cook until deep golden brown and crisp, about 3 minutes per side. Transfer to a rack set on a baking sheet and sprinkle lightly with salt.
Stir the parsley into the beans and divide among plates; drizzle with the remaining 2 tablespoons olive oil. Add the chicken to the plates. Serve with lemon wedges.