Preheat your oven to 350° F. Line a baking sheet with parchment and set aside.
In a medium bowl, combine grated cheese, butter, flour, thyme and 1/4 teaspoon salt. Mix using a spoon or your fingers until it resembles a coarse meal. Pour in the milk and knead the dough using your hands until it forms a shaggy ball. Lightly flour kitchen counter and dump out the dough and any bits. Knead the dough five or six times, until it forms a large disc, adding more milk if needed.
Lightly flour a rolling pin and roll the dough into an 8 x 12-inch rectangle that is 1/4-inch thick. Using a lightly floured sharp knife, cut the dough into 6-inch strips (each about 1/4-inch wide), re-flouring the knife when needed.
Carefully transfer the strips to the parchment-lined baking sheet, spacing them about 1/4-inch apart. Lightly brush each cheese stick with egg wash and sprinkle with remaining 1/4 teaspoon salt.
Place baking sheet in oven; bake until lightly golden brown, about 10 to 12 minutes. Allow cheese sticks to cool on baking sheet and serve at room temperature.