Place the chicken breasts on a cutting board and butterfly them into four thinner pieces. Pat dry and season both sides with salt and pepper.
Heat a large skillet over medium-high heat. Add olive oil.
Add the chicken and cook without moving for 4 minutes. Flip and cook another 4 to 5 minutes until the internal temperature reaches 165°F (74°C). Transfer to a plate.
In the same skillet, melt the butter.
Add chopped onion and sauté for about 3 minutes until softened and lightly golden.
Stir in the garlic and cook for 30 seconds.
Pour in the chicken broth while scraping up the browned bits from the bottom of the pan.
Add thyme and bring to a gentle simmer. Reduce heat to medium-low and simmer for 5 minutes.
While the sauce simmers, prepare the green beans. Heat a separate skillet over medium-high heat.
Melt the butter and add garlic. Cook for 30 seconds until fragrant.
Add green beans, Italian seasoning, red pepper flakes, salt, and pepper. Stir well.
Pour in water and immediately cover the skillet.
Reduce heat to low and cook for 10 to 12 minutes, stirring occasionally.
Remove the lid and squeeze juice from half the lemon over the green beans. Stir and remove from heat.
In a small bowl, whisk cornstarch with 1 tablespoon water until smooth.
Stir the slurry into the chicken sauce and mix until slightly thickened.
Return the chicken to the skillet and spoon the sauce over the top.
Simmer for 2 minutes until heated through.
Serve the Garlic Chicken and Green Beans together with lemon wedges from the remaining half lemon.