Cheddar and chives are a great combination in scones. For light, flaky scones, don't mix the butter in too well.
Heat oven to 400 degrees. Cover a baking sheet with a piece of parchment paper, and set aside. Place the cheddar cheese, chives, and 1 tablespoon flour in a small bowl, and toss to combine; set mixture aside. In a small bowl, combine the 3 lightly beaten eggs and the cream, and set aside.
Using an electric mixer fitted with the paddle attachment, combine remaining 2 cups flour, baking powder, paprika, and salt. Add butter; mix on low speed until butter is in pea-size pieces, 1 to 2 minutes. Add egg-and cream mixture; mix on low speed until just blended. Add cheese mixture; mix until just combined.
Transfer dough to a well-floured surface, and pat into a 9-inch circle. Using a sharp knife dipped in flour, cut circle into 12 wedges. Transfer wedges to prepared baking sheet; brush tops with egg wash. Place in oven; bake, rotating baking sheet halfway through cooking, until outside is crusty and inside is cooked through, 15 to 20 minutes. Transfer to a wire rack to cool.