Adjust oven rack to middle position and heat oven to 300 degrees. Cut 18-inch length foil and fold lengthwise to 8-inch width. Fit foil into length of 13 by 9-inch baking pan, pushing it into corners and up sides of pan; allow excess to over-hang pan edges. Cut 14-inch length foil and fit into the width of the pan in the same manner, perpendicular to the first sheet (if using extra-wide foil, fold second sheet lengthwise to 12-inch width). Grease foil.
Spread almonds, cashews, pepitas, sunflower seeds, flaxseeds, and sesame seeds on aluminum foil-lined rimmed baking sheet. Bake, stirring occasionally, until pale golden and fragrant, 15 to 20 minutes. Transfer nut mixture to food processor and let cool slightly, then pulse until coarsely chopped, about 5 pulses; transfer to a large bowl.
Process ½ cup figs, warm water, maple syrup, egg white, and salt in now empty processor until smooth, about 30 seconds, scraping down sides of bowl as needed. Stir processed fig mixture and remaining ½ cup chopped figs into nut mixture until well combined. Spread mixture in prepared pan and press firmly into even layer using greased metal spatula. Bake until golden brown, 22 to 28 minutes, rotating pan halfway through baking. Do not turn off oven.
Let bars cool in the pan for 15 minutes. Using foil overhang, lift bars out of the pan; transfer to a cutting board and cut into 20 bars. Space bars evenly on parchment paper–lined baking sheet and bake until deep golden brown, 10 to 15 minutes. Let bars cool completely on wire rack, about 1 hour. Serve. (Bars can be stored at room temperature for up to 1 week.)