Place the celery hearts on a large cutting board, trim off any very dark outer ribs and leafy tops with a chef’s knife, and cut each heart into 2 to 3 inch pieces so they fit easily in the pan.​

If the outer ribs feel very stringy, use a vegetable peeler to lightly peel the outside of each piece so the finished celery turns soft and pleasant to bite.​​

Set a large shallow sauté pan or skillet on the stove over medium heat and add 2 tbsp butter and 1 tbsp olive oil to the pan, stirring with a wooden spoon until the butter melts and blends with the oil.​

Add the celery pieces to the pan in a single layer with your hands or tongs and sprinkle ½ tsp kosher salt and ¼ tsp black pepper over the top with measuring spoons.​

Cook the celery for 8 minutes over medium heat, turning the pieces occasionally with tongs so they start to soften and pick up light golden spots on the edges.​

Scatter the sliced shallots into the pan and stir with the wooden spoon so they slip between the celery pieces, then cook for 4 minutes until the shallots look soft and translucent.​

Sprinkle 1 tsp fresh thyme leaves over the celery and shallots with a measuring spoon and stir with the wooden spoon so the herbs coat the vegetables evenly.​

Pour 1 cup broth into the pan using a measuring cup, stirring once with the wooden spoon to loosen any browned bits from the bottom so they mix into the liquid.​

Bring the broth to a gentle simmer over medium heat, then place a lid on the pan and reduce the heat to low so the liquid bubbles softly around the celery.​

Cook covered for 15 minutes, without lifting the lid often, until the celery pieces feel very tender when you pierce one with the tip of the knife or a fork.​

Remove the lid, turn the heat up to medium, and let the celery cook uncovered for 8 minutes, stirring with the wooden spoon every couple of minutes, until the broth reduces to a glossy sauce that just coats the bottom of the pan.​

Use tongs to arrange the braised celery hearts on a serving dish, then spoon the shallots and thyme-scented broth over the top with the wooden spoon, and serve warm alongside roast chicken, beef, or fish.