Pulse all sauce base ingredients in a food processor until it has a pesto-like texture.
In a skillet over medium-high heat, add the sauce base and sauté for 2 minutes. Add almond milk and stir to mix together. Bring to a low simmer for a few minutes until you have a creamy-looking green sauce. Taste and season the sauce with additional salt.
Turn the heat to low. Make 4 shallow wells in the sauce. Crack the eggs into the wells, and cover the skillet with a lid. Let the pan sit for 10 minutes, or until the eggs are cooked.