Whisk together 1 ¼ cup flour with sea salt in a large bowl. Add in cold butter and work together with hands until a coarse, sandy looking meal forms. Add cold water and continue working until the dough melds together. Don't overwork the dough so as to not add too much heat to it.
Shape dough into a square, wrap in plastic wrap, and place in the fridge to chill for at least 30 minutes.
While dough chills, combine peaches, brown sugar, cinnamon, nutmeg, and three tablespoons flour in a large bowl until peaches are coated. Mix in lemon juice.
Melt butter in a large skillet over medium heat and add the filling mixture to the pan. Cook until bubbly and thickened, stirring as necessary (about 10-15 minutes).
Preheat oven to 350 degrees. Pour filling into a square 8 x 8-inch baking pan. Set aside.
Roll out the chilled dough and cut into 8 one-inch strips. Lay the strips over top of peach filling in a lattice or criss-cross pattern.
Brush the dough with beaten egg then sprinkle with white sugar.
Bake cobbler for about 45 minutes or until crust is golden. Allow it to cool and set for at least another 45 minutes then serve.