Pound the chicken thin using a kitchen mallet. Alternatively, slice the chicken crosswise using a sharp knife so that each cutlet becomes 2 thinner cutlets.
Place flour, salt and pepper in a shallow bowl. Whisk to combine.
In a large nonstick skillet over medium heat, melt 2 Tbsp. of butter with 3 Tbsp. of olive oil.
Dredge the chicken cutlets in the flour mixture on both sides. Cook in the skillet until golden brown. Remove onto a separate plate.
Wipe any brown bits off the bottom of the pan and add the remaining 1 Tbsp. of olive oil. Put the shallots and garlic into the pan and saute until softened and sizzling.
Pour the wine into the pan and simmer until the wine is reduced by half.
Stir lemon juice, chicken broth and capers into the pan. Bring to a boil. Cook over medium-high heat until the mixture reduces and begins to thicken, about 3-4 minutes.
Lower the heat. Swirl in the remaining 2 Tbsp. butter. Add the chicken back into the pan and simmer 2-3 more minutes. Garnish with extra lemon slices and fresh parsley. Serve warm.