Melt 2 tablespoons butter in a skillet until foaming subsides. Add shallots and a pinch of kosher salt, and cook over medium-high heat until shallots have begun to soften, about 5 minutes. Reduce heat to low and continue cooking, stirring often, until shallots are soft and brown, about 25 minutes. Deglaze the pan with 1 tablespoon water, scraping up brown bits from bottom and sides of the pan, and stir for an additional minute to allow the water to evaporate. Remove from heat and set aside.
Place prepped potatoes in a large pot and add enough cool water to cover by 1-inch. Add a pinch of kosher salt and bring to a boil. Cook for 15-20 minutes, until potatoes are soft and a knife can be inserted into the center of the wedges easily. Drain potatoes into a colander.
While potatoes are cooking, melt remaining 6 tablespoons butter. Mash drained potatoes to desired consistency (using a hand masher, potato ricer, food mill, or a mixer; I use a ricer for ultra-smooth potatoes). Stir in melted butter, shredded cheddar, 1/2 teaspoon kosher salt, and 1/2 teaspoon freshly ground pepper. Add enough warmed half and half to reach your desired texture (I almost always use the full cup). Fold in caramelized shallots and thyme, and season to taste with additional salt and pepper.
For best flavor, choose an extra-sharp white cheddar for these mashed potatoes. I like to top the potatoes with extra caramelized shallots as a garnish. To do so, add an additional 2 sliced shallots and 1 tablespoon butter to the skillet when caramelizing.