In a medium casserole pot, heat the butter and olive oil on medium heat until lightly foaming.
Add the chicken and cook until lightly brown on both sides, about 5 minutes.
Add salt and pepper, herbes de provence and garlic.
Cover pot and allow chicken to cook for about 10-12 more minutes or more if using bone-in. You want the chicken to be cooked through with clear juices running if pricked with a fork. As the chicken cooks, baste the chicken with its butter several times.
When chicken is done, remove to a plate and keep warm.
Mash the garlic in the casserole with the back of a spoon.
Add the 1/2c of white wine and reduce it by about half over high heat while scraping the bottom of the pot.
In another small pot, beat the egg yolks until they are thick and sticky over medium heat. Beat in lemon juice and 1Tb of white wine.
Next, beat in the casserole liquid very slowly, a few tablespoons at a time. This will create a thick and gorgeous hollondaise sauce.
Continue to beat the sauce over low heat until it is thick and creamy.
Plate the reserved chicken and pour the hollondaise over. Top with freshly chopped parsely and/or basil.