Peel the potato and cut into 1-inch chunks, then place in a pot of cold salted water and bring to a boil until fork-tender.
Season the chicken breast evenly on both sides with the dried oregano, half the sea salt, and half the black pepper.
Heat the olive oil in a skillet over medium-high heat and sear the chicken for 5-7 minutes per side until the internal temperature reaches 165°F.
While the chicken cooks, steam the broccoli florets in a basket over boiling water for 4-5 minutes until bright green and tender-crisp.
Drain the cooked potatoes and return them to the warm pot, then add the Greek yogurt, minced garlic, and remaining salt and pepper.
Mash the potato mixture until smooth and velvety, then stir in the fresh chopped parsley.
Serve the sliced herb chicken over the garlic mashed potatoes with the steamed broccoli on the side.