These next-level breakfast sandwiches get their perfect salty-sweet balance from the addition of peach jam. If you don't have ripe peaches, try the recipe with plums or apricots. Slideshow: More Breakfast Recipes
Bring a medium saucepan of water to a boil. Fill a bowl with ice water. Using a sharp paring knife, mark an X on the bottom of each peach. Add the peaches to the saucepan and blanch until the skins start to peel away, 1 to 2 minutes. Transfer to the ice bath and let cool completely. Peel, pit and thinly slice the peaches. Wipe out the saucepan.
Add the sugar, lemon juice, butter, ginger and 1/3 cup of water to the saucepan and simmer over moderate heat, stirring occasionally, until the sugar has dissolved, about 5 minutes. Stir in the pectin and peaches and bring to a boil. Cook, stirring occasionally, until the peaches begin to soften, about 2 minutes. Transfer the jam to a heatproof jar and let cool completely at room temperature. Cover and refrigerate.
Preheat the oven to 350° and line a baking sheet with parchment paper. In a large bowl, whisk the 2 1/2 cups of flour with the sugar, baking powder and salt. Scatter the cubed butter and lard over the dry ingredients and pinch the fat into the flour with your fingers until the mixture resembles very coarse crumbs, with some pieces the size of small peas. Stir in the buttermilk until a shaggy dough forms.
Turn the dough out onto a lightly floured work surface and knead gently until it just comes together. Pat into a 1-inch-thick round. Using a 3-inch biscuit cutter, stamp out 3 biscuits. Gather the scraps and cut out one more biscuit. Arrange the 4 biscuits about 2 inches apart on the prepared baking sheet and brush the tops with the beaten egg. Bake for about 40 minutes, until golden brown. Transfer to a rack and let cool.
In a large nonstick skillet, melt 1/2 tablespoon of the butter. Working in 2 batches, cook the country ham over moderately high heat, turning once, until golden and crisp around the edges, about 1 minute per batch. Transfer to a plate and wipe out the skillet. In a medium bowl, beat the eggs and season with salt and pepper. Melt the remaining 1/2 tablespoon of butter in the skillet. Add the eggs and cook over low heat, stirring occasionally, until almost set, about 4 minutes. Sprinkle with the cheese, then fold the omelet in half. Cook until set in the center, 1 to 2 minutes longer. Transfer the omelet to a work surface and cut into 4 equal pieces. Split the biscuits and top each bottom half with a piece of omelet and a few slices of ham. Drizzle with the peach jam, close the sandwiches and serve immediately.
The jam can be refrigerated for up to 2 weeks.