In a food processor, combine the artichoke hearts, garlic, 1/4 teaspoon of the salt, the pepper, and olive oil and pulse several times to finely chop; do not puree. Remove from the processor and set aside.
Combine the yeast, sugar, and water in the bowl of stand mixer fitted with a dough hook. Beat on low speed for 1 minute. Add the artichoke mixture, the remaining 1 teaspoon salt, and the flour. Beat until all of the flour is incorporated, about 1 minute. Then beat on medium speed until the dough forms a ball, leaves the sides of the bowl, and starts to climb up the dough hook.
Remove the dough from the bowl. Using your hands, form the dough into a smooth ball. Lightly oil a bowl with the vegetable oil. Place the dough in the bowl and turn it to oil all sides. Cover with plastic wrap and set aside in a warm, draft free place until doubled in size, about 1 hour.
Line two baking sheets with parchment paper.
Remove the dough from the bowl and invert it onto a lightly floured work surface. With your fingers, pat the dough into a rectangle about 3/4 inch thick. Divide the dough into 4 equal portions and form into flattened balls. Using your hands, roll two of the balls of dough into two thick ropes, each 1 inch in diameter and 22 inches long. Join the ropes together at one end and pinch together. Braid the ropes together, lapping each one over the other as you go, then pinch the other ends together. Tuck the pinched ends underneath the braided bread. Place the bread on one of the prepared baking sheets and form it into the shape of a staff (like a candy cane).
Divide the remaining 2 balls of dough in half. You now have four pieces. Using your hands, roll each piece into a thick rope, each about 1 inch in diameter and about 16 inches long. Pinch 2 of the ropes together at one end and braide them together, lapping one over the other as you go. Pinch the other ends together. Repeat the process with the remaining two ropes. Tuck the pinched ends together and tuck them underneath each loaf.
Arrange the 2 braids in the form of a cross on the second baking sheet, gently pulling the lengthwise braid to make it about two inches longer and gently compressing the cross braid to make it fit across the narrow part of the pan.
Cover both breads with plastic wrap and set aside in a warm, draft-free place until doubled in size, about 1 hour.
Preheat the oven to 350 degrees F.
With a pastry brush, brush the beaten egg evenly over the breads. Bake until they are lightly brown and sound hollow when tapped, about 30 minutes. Let cool on a wire rack.