Pat your chicken breasts dry and season both sides generously with salt and pepper. Heat the olive oil in your skillet over medium-high heat until it shimmers. Add the chicken and sear for about 5 to 6 minutes per side until golden and cooked through, then transfer it to a plate.
Lower the heat slightly and add the butter to the same skillet. Let it melt and bubble, swirling occasionally, until it turns a deep amber color and smells toasty and nutty. This usually takes just a couple of minutes, so keep a close eye on it the whole time.
Add your sage leaves directly into the browned butter and let them sizzle for about 30 seconds until they turn crispy and fragrant. Stir in the minced garlic right after and cook it for another 30 seconds, just until it turns golden and aromatic.
Pour in the white wine and use your wooden spoon to scrape up any browned bits stuck to the bottom of the pan. Let the sauce bubble and reduce slightly for a minute or two, then stir in the fresh lemon juice to balance out the richness.
Nestle the seared chicken breasts back into the skillet, spooning the brown butter sage sauce generously over each piece. Let everything warm through together for a minute, then plate it up while the sauce is still glossy and hot.