Bring a large pot of water to a boil. Add chard and cook until bright green and limp, about 1 minute. Transfer the chard to a strainer and press on it to extract as much liquid as possible. Transfer to a cutting board and finely chop. You should have 1 cup; reserve any extra for another use.
Whisk all-purpose flour, whole-wheat flour, yeast and salt in a large bowl. Combine buttermilk and ¾ cup water in a 2-cup glass measuring cup and microwave on High until the liquid reaches 110°F, about 1 minute. (The buttermilk may separate but it's OK.) Combine the buttermilk mixture with the chard and oil in a medium bowl. Add to the dry ingredients, mixing with a wooden spoon or with a paddle attachment in a stand mixer, until you have a moist, shaggy dough.
Turn the dough out onto a lightly floured surface and knead, adding as little flour as necessary, until the dough is no longer sticky, holds together and springs back like a pillow when you poke it with your knuckle, 3 to 5 minutes. (If using a stand mixer, switch to the dough hook and knead, adding flour as necessary, until smooth and elastic, 3 to 5 minutes.) Transfer the dough to an oiled bowl, cover with plastic wrap, and let rise in a warm spot until doubled in bulk, about 2 hours.
Position racks in center and upper third of oven; preheat to 400°F. Line 2 large baking sheets with parchment paper.
Roll out the dough on a lightly floured surface into a 12-inch square. Cut into twelve 1-inch-wide strips. Cut the strips in half to get 24. To shape breadsticks: Cut each strip in half lengthwise. Placing 2 strips together and grasping the ends, twist into a 6- to 8-inch-long breadstick. Place on one of the prepared pans. Repeat with the remaining dough to make 24 breadsticks, placing them about ½ inch apart.
Lightly beat egg and the remaining 1 tablespoon water in a small bowl. Brush the breadsticks with the egg wash and sprinkle with seeds and flaky salt.
Bake, switching the pans from top to bottom and back to front halfway through, until lightly browned, 15 to 20 minutes.
To make ahead: Refrigerate dough (Steps 1-3) overnight. Bring to room temperature before continuing with Step 4 (about 1 hour).
94 calories;2 g fat(0 g sat); 2 g fiber; 17 g carbohydrates; 3 g protein; 33 mcg folate; 8 mg cholesterol; 1 g sugars; 0 g added sugars; 852 IU vitamin A; 3 mg vitamin C; 24 mg calcium; 2 mg iron; 193 mg sodium; 124 mg potassium