Combine the cheese and arugula in a bowl. Cut a deep 2- inch wide pocket in the thickest part of the chicken breast. Stuff with the arugula mixture. Brush with olive oil and season with salt and cajun blend.
Preheat grill to high and brush the grates with vegetable oil. Grill the chicken until blackened and a thermometer inserted into the thickest part reads 155 degrees. ( 8-10 minutes per side) Transfer to a cutting board and let rest.
Meanwhile, heat 1tbs olive oil in a skillet over high heat. Add the lima beans, squash and corn, season with salt and cook until the squash is just tender ( about 3 minutes) Add the tomatoes and cook 2 minutes more. Remove from heat stir in the lime juice. Slice each chicken breast in half and serve with the succotash