Mix 1 1/2 cups warm water (110 degrees F to 115 degrees F), the yeast and agave in a bowl. Set aside 5 minutes.
In a food processor fitted with the dough blade, pulse the whole-wheat flour, 2 cups all-purpose flour and the sea salt. Add 1 egg and the yeast mixture and pulse in 10-second intervals until the dough comes together, about 3 pulses. Turn out onto a floured board. Knead 10 to 12 times, adding up to 3/4 cup more flour if the dough is too sticky. Cut the dough into 4 equal portions; cover with a clean towel.
On a floured board, roll out one piece of dough into a 14-by-11-inch rectangle, 1/4 inch thick, with a shorter side facing you. Spread 2 1/2 tablespoons cream cheese across the middle of the dough; fold the bottom third of the dough over the cheese, then spread another 2 1/2 tablespoons cream cheese and 2 tablespoons pesto over the folded part. Fold the top third of the dough over the pesto. Reroll the dough into a 14-by-11-inch rectangle. Cut into seven 11-by-2-inch strips.
One at a time, pinch the cut long edges of each strip together and roll into a 12-inch rope with your hands. Transfer the rope to a baking sheet lined with parchment paper. Grab the ends and bring them toward each other, forming a U shape, then cross the left end over the right end to make a pretzel shape. Cross the left end over the right again to make a twist in the middle.
Repeat with the remaining 3 pieces of dough and the remaining cream cheese and pesto, spacing the pretzels 3 inches apart. Set aside for 10 to 15 minutes; preheat the oven to 475 degrees F.
Beat the remaining egg in a bowl. Brush the pretzels with the egg; sprinkle with coarse sea salt and dried basil. Bake until browned, 16 to 18 minutes, checking on the pretzels every 5 minutes to avoid overbaking.
Photograph by Victor Schrager
Recipe courtesy Guy Fieri for Food Network Magazine