Preheat oven to 400°F and line a sheet pan with parchment paper.
Whisk heavy cream, egg, and dry milk powder together in a liquid measuring cup.
In a food processor or blender, add 20 chives, ½ cup dill fronds, and ½ cup parsley leaves. Pulse until finely chopped.
Add flour, baking powder, and salt to food processor/blender and pulse until well combined with the herbs.
Add cheese and pulse until well combined.
Cut cold stick of butter into small cubes and add to food processor/blender a few cubes at a time. Pulse and add more butter until only a few pea-sized pieces of butter remain.
Transfer mixture from the food processor/blender to a large bowl and make a well in the center. Pour the wet ingredients into middle of well and slowly incorporate the dry mixture with a fork until a shaggy dough forms.
Lightly knead dough unti it comes together. Turn dough out onto a floured workspace and pat into a large rectangle. Use a floured, 2-inch-wide heart-shaped cookie cutter to stamp out about 20-24 hearts.
On the prepared sheet pan, arrange 4 of the hearts in a four-leaf clover shape with the points of the hearts meeting in the center. Repeat with the remaining hearts for a total of 5-6 clovers, spacing them evenly apart. Freeze the pan for 15 minutes.
Remove pan from freezer and brush the top of the clovers with heavy cream. Sprinkle with flaky salt. Bake the scones for 20-25 minutes, or until golden brown.
Meanwhile, pulse the remaining 10 chives, ¼ cup dill fronds, and ¼ cup parsley leaves in a food processor or blender until finely chopped. Add the stick of room-temperature butter and pulse until the mixture is light green and well combined.
Serve the scones warm with the herb butter.