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For the biscuits:
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Cut cold butter into 1/2-inch cubes. Beat buttermilk and one egg lightly in a small bowl. Toss grated cheese and chives with 1 tablespoon of extra flour to coat. Refrigerate butter, egg mixture, and cheese until needed. Preheat oven to 425u00b0F with rack in middle and line a half sheet pan with parchment. Set up a stand mixer with whisk and have paddle attachment ready. Clean and lightly flour your workspace and a rolling pin. Beat remaining egg with a tablespoon of water and set aside.
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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, 1-2 minutes. Switch to 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.
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Turn mixture out onto prepared surface and press it all 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. 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. Place biscuits gently on prepared baking sheet, brush liberally with reserved egg wash, and bake 20-25 minutes, rotating the pan once. Biscuits are done when risen, fluffy, and just golden. Cool a minute or two on pan then transfer to a wire rack to cool completely.
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For the garlic chive butter:
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Using a fork, or in a food processor or electric mixer, mix garlic salt into softened butter, and the chives. Taste and add more garlic salt as desired. Refrigerate butter in a small air tight container until firm.
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To assemble the sandwiches:
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Whisk egg 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 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 3-5 minutes. Set biscuits face up on plates or a cutting board and spread with a little garlic chive butter.
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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 1 slice on each biscuit half (2 slices per sandwich). Add remaining garlic chive butter to skillet. Once melted, pour in eggs and cook until beginning to set. Use a spatula to divide roughly into 4 rectangles, 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).
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Notes:
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1. Prep and cook times INCLUDE biscuit preparation. If you've never made biscuits before, allow a little extra time to work slowly.
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2. Biscuit recipe yields 8 biscuits, twice as many as needed. Freeze or eat the extras, or buy more sandwich ingredients to serve a crowd!
