Preheat your oven to 350°F (175°C). Lightly grease a 9x9‑inch baking dish (or similar 2‑quart dish) with butter or nonstick spray. A glass or ceramic dish works well.

If using fresh peaches, peel them if desired (peel all or just some; the skins soften as they bake). Slice the peaches into about 1/2‑inch wedges until you have about 6 cups. Place the sliced peaches in a large mixing bowl.

To the bowl with peaches, add the lemon juice, vanilla extract, granulated sugar, brown sugar, cornstarch, ground cinnamon, ground nutmeg (if using), and a pinch of salt. Gently toss until the peaches are evenly coated and the mixture looks glossy and slightly syrupy. If it seems very dry, add 1 more tablespoon lemon juice or a small splash of water. Pour the peach mixture into the prepared baking dish and spread into an even layer.

In a separate medium bowl, combine the oats, flour, brown sugar, granulated sugar, ground cinnamon, and salt. Stir to blend evenly. Add the cold butter cubes and cut them into the dry ingredients using a pastry cutter, two forks, or your fingers until the mixture looks like coarse crumbs with some pea‑sized bits. Stir in the chopped nuts, if using. The mixture should clump when squeezed but break apart easily.

Sprinkle the topping evenly over the peaches, covering the surface from corner to corner. Place the baking dish on a parchment‑lined baking sheet to catch any bubbling juices. Bake for 35–45 minutes, until the topping is deep golden brown and the peach filling is bubbling around the edges. If the top browns too quickly, tent loosely with foil for the last 10 minutes of baking.

Remove the peach crisp from the oven and let it cool for at least 15–20 minutes so the juices can thicken. Serve warm with vanilla ice cream, whipped cream, or Greek yogurt.