Start by peeling and cutting your Yukon Gold potatoes into evenly sized chunks. Place the potatoes in a large pot and cover them with cold water. Add a pinch of salt to the water and bring it to a boil over high heat.
Once the water is boiling, reduce the heat to medium and let the potatoes simmer until fork-tender, about 15-20 minutes. Drain the potatoes and set them aside.
While the potatoes are cooking, steam the broccoli florets until they are bright green and tender but still slightly crisp, about 4-5 minutes. Use a steamer basket over boiling water or microwave them in a covered dish with a splash of water.
In a small pan, melt 2 tablespoons of butter over medium heat. Add the minced garlic and sauté for about 1 minute until fragrant but not browned.
Return the drained potatoes to the pot or a large mixing bowl. Add the garlic butter, remaining 2 tablespoons of butter, and the warm milk. Using a potato masher or electric mixer, mash until smooth and creamy. Be careful not to overmix.
Fold in the steamed broccoli florets and shredded cheddar cheese gently into the mashed potatoes. The warmth of the potatoes will melt the cheese slightly.
Season the mashed potatoes with salt, freshly ground black pepper, and, if using, smoked paprika. Mix well and adjust seasoning to taste. Serve warm.