Cook broccoli florets and carrot matchsticks, in a large pot of boiling salted water until crisp-tender, about 1 minute. Transfer to a fine-mesh sieve with a spider or slotted spoon
Gently lower 4 large eggs into pot of boiling water. Cover and cook, adjusting heat as needed to maintain a boil, 8 minutes; transfer to sieve. Let vegetables and eggs cool
Place rice noodles in boiling water, cover pot, and remove from heat. Let sit, stirring occasionally, until tender, 10–15 minutes. Drain and rinse under cold running water until cool. Shake off excess water; set aside
Meanwhile, bring light brown sugar and soy sauce to a boil in a medium skillet over medium-high heat, stirring to dissolve sugar. Cook, stirring often, until mixture is thickened and syrupy, about 3 minutes. Measure out 1/3 cup of mixture in a heatproof measuring glass and pour into a medium bowl
Reduce heat to medium and tofu and vegetable oil to remaining soy syrup in skillet. Cook, turning occasionally, until tofu absorbs sauce, about 4 minutes
Vigorously whisk garlic, peanut butter, lime juice, 1 tbsp sambal oelek, salt to taste, and 1/3 cup warm water into soy syrup in bowl, adding more warm water by the tablespoonful if needed, until peanut sauce is creamy (it should be the consistency of ranch dressing)
Peel eggs and cut into quarters. Spread peanut sauce on a platter or in a large shallow bowl; top with reserved noodles. Arrange eggs, broccoli, carrot, tofu, avocado, cucumbers, and peanuts over the top. Toss and serve with more sambal oelek