For the Taquitos: Heat 1 teaspoon oil in 12-inch nonstick skillet over medium-high heat until just smoking. Add beef and cook, breaking up pieces with spoon, until no longer pink, about 5 minutes. Drain beef in colander; set aside. Mash beans to past with potato masher or fork; set aside.
Heat 1 tablespoon oil in now-empty skillet over medium heat until shimmering. Add onion and cook until lightly browned, about 5 minutes. Stir in jalapeños, garlic, cumin, and chili powder and cook until fragrant, about 30 seconds. Stir in mashed beans, tomato sauce, water, cilantro, 1/2 teaspoon salt, and 1/2 teaspoon pepper. Stir in drained beef and cook, stirring occasionally, until mixture has thickened and begins to sizzle, about 10 minutes. Season with salt and pepper to taste. Transfer to bowl; set aside to cool, about 20 minutes.
Line rimmed baking sheet with parchment paper. Set wire rack in second rimmed baking sheet. Wrap 6 tortillas in clean, damp kitchen towel; place on plate; and cover plate tightly with plastic wrap. Microwave until hot and pliable, about 90 seconds.Working with 1 tortilla at a time, brush edges of top half with egg. Place row of 3 level tablespoons filling across lower half of tortilla. Fold bottom of tortilla up and over filling, then pull back on tortilla to tighten it around filling. Roll tightly and place seam side down on lined baking sheet. Cover with second clean damp towel. Microwave remaining 6 tortillas and repeat with remaining filling.
Adjust oven rack to middle position and heat oven to 200 degrees. Heat remaining 1 cup oil in clean, dry 12-inch nonstick skillet over medium-high heat until 350 degrees. Using tongs, place 6 taquitos, seam side down, in oil and fry until golden, about 5 minutes. Flip and fry until second side is golden, about 3 minutes longer. Transfer to wire rack and place in oven to keep warm. Repeat with remaining 6 taquitos.
For the Avocado Sauce: Meanwhile, combine avocados, sour cream, water, lime juice, and cilantro in bowl and mash with potato masher or fork until smooth. Season with salt and pepper to taste. Cover with plastic until ready to serve. Serve taquitos with sauce.
Notes: To make ahead - Taquitos can be covered with a damp towel, wrapped tightly in plastic, and refrigerated for up to 24 hours before frying.
