Combine the almonds and the soaking water in a medium bowl. Cover and soak 8–12 hours at room temperature or in the refrigerator until the nuts are swollen and break easily when pinched.
Drain and discard the liquid. For a whiter, milder milk, pinch off and discard the almond skins; leaving skins on yields a deeper almond flavor.
Transfer the drained nuts to a blender and add the blending water. Blend on high until very fine and frothy, 45–60 seconds; the mixture should look uniformly opaque with no visible grit.
Strain through a nut milk bag or a fine-mesh sieve lined with a double layer of cheesecloth set over a bowl. Squeeze firmly until the pulp feels nearly dry, 2–3 minutes.
Pour the almond milk into a clean bottle or jar. Chill at least 1 hour, then refrigerate up to 4–5 days; shake before each use as natural separation will occur. Reserve the squeezed almond pulp for baking or smoothies if desired.