Cheese biscuits

1. Cut cold butter into 1/2″ cubes (freeze it for a few minutes first if kitchen is warm). Beat buttermilk and extra large egg lightly in a liquid measure or small bowl. Toss grated cheese and chives with about 1 tablespoon of flour to coat. Refrigerate butter, egg mixture, and cheese until needed.
2. Preheat oven to 425 degrees (F) with rack in middle and line a half sheet pan with parchment or a silicone mat. Set up a stand mixer with whisk and have paddle attachment ready. Clean and lightly flour a large board or countertop workspace and lightly flour a rolling pin. Beat remaining egg with a tablespoon of water and set aside.
3. In mixer, whisk flour, baking powder, and salt on low to combine. Add cold butter and whisk on medium low until butter is in pea size chunks, about 1 to 2 minutes. Switch to the paddle attachment and add buttermilk mixture all at once and mix on low until barely combined. With mixer running, add cheese and chives and mix briefly until a cohesive dough has just formed. If there are some loose crumbs or flour in the bottom of the bowl, they can be incorporated while kneading.
4. Turn mixture out onto prepared floured surface and press it all, including any loose or dry pieces, into a single mass. Knead mixture about 6 to 10 times, each time pressing it down with the heel of your hand then folding the farthest edge back toward you, until combined well but still cold. Resist the temptation to keep kneading, or you may end up with flat, dense biscuits. Press dough into a short log, then roll out to a thick 10 by 5 inch rectangle. Using a knife or board scraper, cut straight down to divide dough in half lengthwise, then crosswise into quarters.
5. Place 8 biscuits gently on prepared baking sheet, brush liberally with reserved egg wash, and bake 20 to 25 minutes, rotating the pan once if your oven cooks unevenly in front and back. Biscuits are done when risen and fluffy and just golden. Cool a minute or two on pan then transfer to a wire rack to cool completely.

Garlic chive butter

1. Make sure butter is very soft. Using a fork, or in a food processor or electric mixer, mix in 1/2 teaspoon garlic salt to start, and the chives. Taste (on a scrap of bread, if you have one handy) and add more garlic salt as desired. Refrigerate butter in a small air tight container until firm, or place in a rough log shape at one end of a large sheet of wax paper on top of a layer of foil (kind of like this). Roll into a log, twisting or pinching the ends as you go, to form a cylinder.

Sandwich assembly

1. Multiply ingredient quantities for as many sandwiches as you need. Whisk egg(s) well with a splash of milk plus pepper and set aside. Have all other ingredients out and ready to go. While heating a dry nonstick or well seasoned cast iron skillet over medium, halve biscuit(s) and spread each half generously with mayonnaise. Toast cut side down in hot pan until browned and warm, about 3 to 5 minutes. Set biscuits face up on plates or a cutting board and spread with a little garlic chive butter.
2. Turn heat down slightly and cook prosciutto slices for a minute or two on each side in a single layer, until slightly rendered and warm, but still pliable. Arrange one slice on each biscuit half (two slices per sandwich). Add remaining garlic chive butter to skillet. Once melted, pour in egg(s) and cook until beginning to set. Use a spatula to divide roughly to the size of each sandwich, then flip pieces over and cook until done. Salt lightly if desired, keeping in mind the garlic salt in the butter. Place fried eggs on each biscuit bottom, stacking remaining prosciutto and biscuit tops to form sandwiches. Serve immediately (keep warm on a covered tray or plate in a low oven if you’re making a lot of sandwiches).