Season the chicken with salt and pepper, and rub it with the basil pesto.
Heat the olive oil in a large nonstick skillet set over medium-high heat. Add the chicken to the heated oil and cook for 5 to 6 minutes on each side or until thoroughly cooked.
In the meantime, prepare the sauce by combining flour, garlic, basil pesto, evaporated milk, salt, and pepper in a mixing cup or bowl; whisk until thoroughly combined and set aside.
Remove the chicken from the skillet and set it aside.
Return the skillet to the heat and stir in the tomatoes. If the pan is dry, add a little more olive oil.
Cook the tomatoes over medium-high heat for about 1 minute or until just beginning to soften. Add the prepared pesto sauce to the skillet and bring to a boil.
Place the chicken back in the skillet and cook for 2 to 3 minutes, or until heated through.
Remove from heat and garnish with cheese and basil ribbons.