Femtosecond mode-locked lasers are the workhorse of modern photonics, enabling optical atomic clocks, frequency combs, precision spectroscopy, and ultrafast signal processing. Bringing them onto nanophotonic chips promises miniaturized ultrafast systems, yet combining femtosecond pulse generation with electrical pumping remains a major challenge, rooted in a trade-off between the speed and energy efficiency of mode-locking mechanisms. Kerr-based mechanisms offer nearly instantaneous response but typically require high intracavity peak powers that are difficult to achieve with nanophotonic-compatible semiconductor gain. Conversely, semiconductor saturable absorbers enable strong nonlinear responses at low energies but are limited by their slow carrier dynamics. Here we break this trade-off by introducing quadratic-soliton mode-locking based on a quadratic nonlinear interferometer (QNLI): an effective saturable absorber built from cascaded second-order nonlinearity in periodically poled lithium niobate nanophotonics, with a measured sub-20 fs response time. Through hybrid integration with an electrically pumped semiconductor gain element, we demonstrate a self-starting quadratic-soliton mode-locked laser with femtojoule pulse energies, operating in single- and multiple-soliton states. Quadratic solitons arise through the QNLI's dissipative and dispersive nonlinear response, which we confirm with a modified complex Ginzburg-Landau equation in agreement with measurements. By combining soliton formation with intracavity electro-optic modulation, we generate 750 fs pulses with flat-top spectra. Our engineerable, ultra-energy-efficient, electrically powered laser provides the critical source for fully integrated ultrafast photonic circuits in lithium niobate nanophotonics that natively offers electro-optic control, quadratic wavelength conversion, pulse compression, and spectral broadening.