Steam the green beans until crisp-tender (or until they reach your desired level of doneness).
In a large skillet, heat the olive oil and butter over medium heat. Add the chicken, season with salt and pepper, and sauté, stirring frequently, until the chicken is cooked through and no longer pink (we're not going for browned chicken in this recipe; be careful not to overcook the chicken so it is not dry).
Add the garlic and sauté for another 1 to 2 minutes, stirring frequently. Note: keep an eye on the heat and lower it, if necessary, to make sure that the garlic doesn't burn.
Add the steamed green beans; sauté, stirring frequently, for another minute.
Turn off the heat and add the tomatoes; stir to combine. Taste and add salt and pepper, if needed.
Best when served immediately. Sprinkle each serving with grated Parmesan cheese, if desired.