The phrase
"vigorous simmer" means to cook something over medium to high heat with a steady, bubbling motion in the liquid, but not boiling too vigorously.
Full definition
Reduce heat to maintain
a vigorous simmer.
Add the broth, water, reserved mushroom soaking liquid, mirin, soy sauce, vinegar, and chile paste, and bring to
a vigorous simmer over high heat.
Bring to a boil, then reduce heat to
a vigorous simmer, skimming off any white foam that rises to the top.
Return to high heat, bring to
a vigorous simmer, and cook stirring occasionally, until sauce is thickened and measures 1 1/4 cups, about 5 minutes.
Cover and reduce the heat to a low boil or
vigorous simmer.
Stir in squash and 2 1/2 cups broth and bring to
a vigorous simmer, about 5 to 7 minutes.
Bring to a boil, then reduce to
a vigorous simmer and let cook until liquid reduces by half and thickens up a bit, 20 to 30 minutes.