Arrange the bread and almonds on a baking sheet, and bake, flipping the bread once, until the bread is toasted and crunchy and the almonds are golden, about 10 minutes. Set aside to cool.
Meanwhile, put the oil, vinegar, orange zest and juice and 1/4 teaspoon salt in a jar or other container with a lid. Secure the lid and shake until combined. (The vinaigrette can be made a day ahead; shake well before serving.)
Put an avocado half on each slice of toast and mash and spread the avocado with a fork to cover, season with salt. Quarter each piece of toast on the diagonal, and arrange the pieces with the points facing out to make a square on 4 plates.
Toss the arugula, Cheddar, ham, radishes, toasted almonds and vinaigrette in a medium bowl. Taste and adjust seasoning with salt if necessary. Put a mound of salad on top of each avocado-toast square. Top with a few grinds of pepper.
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