Set a large skillet over medium-high heat and add in olive oil and butter.
Add eggs and Parmesan cheese to a shallow bowl and beat/mix until well combined.
Add flour, salt, and pepper in a separate shallow plate and stir to combine.
Working with one chicken filet at a time, coat both sides of each chicken filet with flour; then dip in the egg mixture to coat on all sides. Allow excess egg mixture to drip off.
Add filets to the heated oil and cook for about 4 to 5 minutes per side, or until cooked through and golden brown. If chicken is browning too quickly, but not cooking through on the inside, lower the heat.
Remove chicken filets from skillet and transfer to a paper towel lined plate. Cover with a paper towel and set aside.
Set skillet back over medium-high heat. Add garlic to skillet and cook for 15 seconds, or until fragrant. Stir so it doesn't burn.
Slowly whisk in chicken broth, scraping up any browned bits from the bottom of the skillet.
Continue to cook for 4 minutes, or until liquid is reduced.
Lower heat to medium-low and whisk in heavy cream and 2 tablespoons parsley.
Continue to cook for 2 minutes, or until thickened.
Whisk in lemon juice; whisk until completely incorporated and continue to cook for 2 minutes. It's important to whisk continuously so the cream does not curdle when you add in the lemon juice.
Remove from heat.
Add chicken pieces back into the pan; spoon sauce over chicken, garnish with parsley, and serve.