In the bowl of a stand mixer fitted with the paddle attachment, stir together 2 1/2 cups flour, sugar, yeast, and salt; add 3 eggs and buttermilk. Beat at medium-low speed for 3 minutes. Turn off mixer; switch to the dough hook attachment. With mixer at low speed, add remaining 1 1/4 cup flour, a few tablespoons at a time as needed, until a soft dough forms and starts to pull away from bowl, about 10 to 12 minutes more. Dough will be slightly sticky.

On a lightly floured surface, turn out dough and shape into a ball. Spray a large bowl with cooking spray. Place dough in bowl, turning to grease top. Cover and let rise in a warm, draft-free place (75°) until doubled in size, about 30 minutes.

Preheat oven to 350°. Spray an 8x4-inch loaf pan with baking spray with flour.

Punch down dough, and turn out onto a lightly floured surface. Roll dough into a 12-inch square. Spread Garden Herb Pesto on dough, leaving a 1-inch border on one side. Starting from side opposite border, roll up dough into a log, pinching seam to seal; place seam side down. Using a serrated knife, cut log in half lengthwise, stopping 1 inch from one end of log. Twist halves together, cut sides facing up. Transfer twisted dough log to prepared pan, tucking ends under to fit. Cover and let rise in a warm, draft-free place (75°) for 20 minutes.

Whisk remaining egg until foamy. Carefully brush dough with egg, avoiding the exposed pesto. Bake until top is golden brown and an instant-read thermometer inserted in center registers 190°, 50 to 55 minutes, covering with foil halfway through baking to prevent excess browning. Let cool in pan on a wire rack for 15 minutes. Remove from pan and let cool completely on a wire rack.