Place the oil in a large sauté pan and set over medium heat. Once the oil shimmers, add the garlic and shallots, and cook, stirring frequently, until golden and crisp, 2 to 3 minutes Watch carefully to prevent burning. Remove the garlic and shallots from the pan using a slotted spoon and transfer to a paper towel–-lined plate to drain.

Add the arbol, ancho, and guajillo chilies and cook until crisp and aromatic, 2 to 3 minutes. Remove the chilies from the pan with the slotted spoon and transfer to the plate to drain.

Decrease the heat to low and wait 3 to 4 minutes to allow the oil to cool slightly. Very carefully place the ginger, tamari, Aleppo pepper, star anise, cinnamon stick, sugar, cardamom, cumin, and garlic into the oil and simmer 10 to 12 minutes to infuse the oil. Strain through a fine mesh strainer and reserve the oil.

Roughly chop the fried shallots and garlic and transfer to a mixing bowl. Finely chop the fried chilies and add them to the mixing bowl. Add the reserved oil and mushroom powder. Taste and adjust the seasoning with salt as necessary. Transfer to an airtight container and keep at room temperature for up to 6 months.

Preheat a grill to medium-high heat.

Place the squash in a large bowl and add the oil, salt, and pepper and toss to evenly coat. Allow any excess oil to drain from the squash and place cut side down onto the hot grill. Cook until the squash has grill marks, 2 to 3 minutes. Turn the squash 90 degrees and continue to cook 2 to 3 minutes. Turn the squash 90 degrees one last time and cook, until the squash is golden brown, 2 to 3 minutes. Remove the squash to a platter and set aside.

Place a sauté pan over high heat and heat 4 minutes. Place the tomatoes in the pan and cook until charred on one side, about 4 minutes. Turn the tomatoes and cook on the other side until charred, about 4 minutes.

Transfer the tomatoes to the platter with the squash. Generously spoon the garlic chili crisp over the vegetables, squeeze the lemon juice over the vegetables, and serve immediately.