Bring the stock to a boil in a small saucepan. Meanwhile, toast the bulgur in a medium skillet over medium heat until fragrant and nutty, about 4 minutes. Stir in the bulgur, lower the heat, and simmer, covered, for 15 minutes. Remove from the heat and set aside for 5 minutes.
Transfer the bulgur to a medium bowl and fluff with a fork. Add 1/4 cup of the olive oil, chopped herbs, pine nuts, 2 teaspoons salt, scallions, and garlic and season with pepper, to taste. Cool slightly and add the egg and feta, and mix to combine. Set the stuffing aside.
Preheat the oven to 450 degrees F.
Lay the hens on a work surface. Starting from the neck of a hen, slip a couple fingers between the skin and breast, and loosen the skin. Take a large spoonful of the stuffing and slip the spoon between the skin and breast. With your free hand, hold the stuffing in place and gently pull the spoon out, leaving the stuffing behind. Press the skin down to evenly distribute the stuffing over the breast. Repeat until the hens are evenly stuffed. Cross the legs of the birds at the breastbone and tie them with a piece of kitchen twine.
Arrange the hens breast-side up on a baking sheet. Brush all over with the remaining 2 tablespoons olive oil and season generously with salt and pepper.
Roast the birds until well browned and crispy, and an instant-read thermometer registers 160 degrees when inserted into the thigh, about 40 minutes.
To serve, split the hens in half lengthwise. Pour the coulis onto a platter and arrange the hens on top. Garnish with the herb sprigs and serve.
Put the tomatoes over 2 gas burners on high and cook, turning occasionally, until charred and peeling. Alternatively, broil the tomatoes until charred. Put the tomatoes in a bowl and cover tightly with plastic wrap. Set aside to steam for 5 minutes. Peel and seed the tomatoes.
Transfer the tomatoes to a blender with the vinegar and garlic and puree until smooth. While the motor is running, drizzle in the oil until incorporated. Season with pepper, to taste.
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