Start by bringing a large pot of salted water to a boil. Add the peeled and cut Yukon Gold potatoes and cook until fork-tender, about 15-20 minutes. Drain and set aside.
While the potatoes cook, heat 1 tablespoon of butter and 1 tablespoon of olive oil in a large skillet over medium-high heat. Pat the chicken thighs dry and season generously with salt and pepper.
Sear the chicken, skin side down first, until golden and crisp, about 5-7 minutes. Flip and cook the other side for another 5 minutes. Remove the chicken from the skillet and set aside on a plate.
Reduce the heat to medium. In the same skillet, add the chopped onion and sauté until translucent, about 3-4 minutes. Add the minced garlic and cook for an additional 1 minute, stirring constantly to avoid burning.
Sprinkle the flour over the onions and garlic. Stir well to combine and cook for 2-3 minutes, allowing the flour to brown slightly — this builds that rich, nutty flavor Mama Lu always talked about.
Slowly whisk in the chicken broth, scraping the browned bits off the bottom of the skillet. Bring the mixture to a gentle simmer, then stir in the 1/2 cup milk or cream. Continue cooking until the gravy thickens, about 5 minutes.
Return the chicken thighs to the skillet, nestling them into the gravy. Cover and simmer on low for 15-20 minutes, or until the chicken is cooked through and tender.
While the chicken simmers, mash the cooked potatoes with 2 tablespoons of butter and 1/4 cup milk or half-and-half. Stir in the shredded cheddar cheese until melted and creamy. Season with salt and pepper to taste.
When the chicken is done, give the gravy a final stir and adjust seasoning if needed. Spoon the cheesy mashed potatoes onto plates and top with a chicken thigh and a generous ladle of garlic gravy.
Garnish with fresh parsley if you like, and serve immediately. Don’t forget to soak up every last bit of that gravy — trust me, it’s worth the extra napkins.